I often think that there are essentially two flavours of theorem in mathematics. Some are direct statements asserting particular truths about important special cases, while others are simple-looking statements asserting more general truths that, rather than merely giving us ‘predictive power’ over a particular locale of the abstract mathematical landscape, give us a true taste of the underlying nature of some object (or commonly, some relation between objects—although that is in itself an object, of course).

Theorems of the first kind, I think, could include the celebrated and very recent Green-Tao theorem that asserts the existence of arbitrarily long arithmetic progressions in the primes, or, to take a random example, the amusingly-named Hairy Ball theorem which tells us a certain property of tangent vector fields on even-dimensional n-spheres. They tell us that something is true, there is a proof given to convince us, and they more or less provide their own motivation for existence. They’re just good things to know. It’s likely these theorems have deeper implications that I’m not aware of, but I think I can rightly claim they are to an extent self-motivating and fairly restricted in scope.

A prime example of the second kind is what is often referred to as the First Isomorphism Theorem.

(In this article I will talk about the theorem in the context of group theory, but it has analogues throughout algebra that apply to other structures such as rings, vector spaces, modules and other even more general objects)

When I first saw this theorem for the first time I felt like I understood it. I mean, it seems simple enough.

Let $\varphi: G \to H$ be a group homomorphism.

Then $\ker(\varphi) \trianglelefteq G$ and $G / \ker(\varphi) \cong \varphi(G)$

I thought, OK, so there’s a homomorphism between two groups and it says that that particular quotient (by the kernel of the homomorphism) is isomorphic to the image of that homomorphism. Also, the kernel (the set of elements that map to the identity) is a normal subgroup of the domain. Wait, so there are two maps going on here—a homomorphism and an isomorphism. And they’re related? Actually, I’m confused now. Who cares that this correspondence exists anyway?

Sometimes it’s stated in a slightly more complicated-looking way where the isomorphism is explicitly defined.

… then there is a group isomorphism $\bar{\varphi}: G / \ker(\varphi) \to \varphi(G)$ such that $\ker(\varphi) g \mapsto \varphi(g)$.

This map $\bar{\varphi}$ is referred to as the induced (or canonical, or something) isomorphism, but is really just extraneous strength that I don’t think is part of the important point of the theorem, so we can safely ignore it for now.

Now, I had two doubts at this point. The first is the question of what this theorem means, and the second is the question of what the point of homomorphisms actually is. Isomorphisms are clearly useful—they tell us when to stop looking. Two things are isomorphic if they’re basically the same thing, and further differences can be considered unimportant facts of the particular representation or implementation details, if you like. But homomorphisms? They seem to preserve *something* but corrupt some of it at the same time.

When I realised the true importance of this theorem and what it really says—what Eugenia Cheng and others might call its ‘moral message’—I wondered why it was called the first isomorphism theorem. Surely it should rather be called the first *homomorphism* theorem, or perhaps since there is really only one of these theorems, the *fundamental homomorphism* theorem. Alas, upon googling briefly I find that I am not as profound as I think I am—it seems to be referred to by some as exactly that.

So what is its moral message?

The fundamental homomorphism theorem (I’ll call it that now) tell us fundamentally, what homomorphisms do. Surprise, surprise!

We can split all possible homomorphisms into essentially two kinds: embeddings and quotients. Embeddings are injective; quotients aren’t.

Embeddings do just what they say on the tin. They map a particular group into another group without blurring the distinction between elements. Examples include maps like $\theta_1: \mathbb{Z} \to \mathbb{R}$ such that $x \mapsto x$ or $\theta_2: \mathbb{Z}_2 \to \mathbb{Z}_6$ such that $x \mapsto 2x$.

You can think of the first group as being ’embedded’ in the second in the sense that there is an actual lookalike copy of it (an isomorphic subgroup) contained within the second. In fact, the image of a homomorphism is always a subgroup (though not necessarily a normal one) of the codomain. This is sometimes included in the statement of the first isomorphism theorem, although it is really a straightforward corollary. In the case of $\theta_1$, the integers are embedded in the reals in the obvious sense, and in the case of $\theta_2$, the integers under addition modulo $2$ are embedded in the integers modulo $6$.

Quotients, on the other hand, do blur the distinction between group elements as they are mapped into the new group. This is clearly more complicated than a simple embedding. Is there an intuitive picture we can use to visualise the image of these kind of homomorphisms in the same way we can with embeddings? Yes, and the fundamental homomorphism theorem tells us how. It says that **images of quotient maps are quotient groups**—in particular, quotients by the kernel. The clue was in the name!

So the theorem is really quite intuitive: it tells us exactly how homomorphisms blur groups as they map them into other groups. If the kernel is trivial, the homomorphism preserves the group entirely and we have an embedding, and if it’s not trivial, it has the effect of mapping it to a subgroup that is isomorphic to the original group ‘quotiented out by’ the kernel. Of course, if we map a group into another group using some arbitrary function, we may end up with some strange subset that isn’t even a group. But the theorem tells us that as long as we have a homomorphism, that is, a function that is compatible with the group operation, the subset we map to will always be a group—and even better, we know exactly what kind of group it will be.

That’s the first question sorted; we now know what the fundamental homomorphism theorem says. How about the second question? What is the point of homomorphisms, anyway?

I think an example will be enlightening here.

Take the determinant map $\det: GL_{n}(\mathbb{R}) \to \mathbb{R}^*$ mapping from the general linear group of degree $n$ (the set of all $n \times n$ real-valued invertible matrices) to the nonzero real numbers. Considering $GL_n(\mathbb{R})$ and $\mathbb{R}^*$ each as groups under multiplication, the determinant is a homomorphism of groups since

$$\det(AB) = \det(A)\det(B) \text{ for } A, B \in M_{n \times n}(\mathbb{R})$$

as is fairly easily proven.

The point of the determinant is that takes a matrix, which is a rather complicated thing, and associates with it a relatively simpler thing, a real number, so that we can deal with that instead. The determinant of a matrix doesn’t capture all of the information about that matrix, but it certainly tells us something important. For example, matrices with nonzero determinant are precisely the invertible matrices, and are the ones that represent injective linear maps. The determinant, geometrically, can be thought of as the scaling factor of the linear transformation the matrix represents.

The fact that it is a homomorphism allows us to freely move between the parallel worlds of matrices and their associated real numbers without running into contradictions. If, say, we want to find the determinant of a product of two matrices, we can compute the product and then find its determinant, or we can just multiply the two determinants as real numbers. The two worlds have a nice correspondence in this way. If our function wasn’t a homomorphism, the correspondence would not be exact and moving between the two could cause some issues.

Despite this niceness, even with a homomorphism something is clearly lost. We can easily find two matrices which have the same determinant, which means we can’t even seem to distinguish between them purely on the basis of their determinants. But this is the tradeoff for simplifying our lives. What we have found is that the determinant is a non-injective homomorphism (a quotient map!) and the fundamental homomorphism theorem will tell us its structure, no questions asked.

The kernel of this particular determinant map is $\{A \in GL_n(\mathbb{R}) : \det(A) = 1\}$, the set of real matrices with determinant $1$ (the multiplicative identity in $\mathbb{R}$). It’s usually known as the special linear group $SL_n(\mathbb{R})$ and it’s the set of linear transformations that preserve volume (and orientation—meaning that they don’t ‘flip’ space).

By the fundamental theorem and the fact that the determinant map is surjective (into the nonzeroreals), we have

$$GL_n(\mathbb{R}) / SL_n(\mathbb{R}) \cong \det(GL_n(\mathbb{R})) = \mathbb{R} \setminus \{0\}$$

and so we essentially have the real numbers (without zero). What the determinant has done is to identify certain matrices that have some deep property in common. The group element corresponding to the real number $3$, for example, is in reality the infinite set $\{A \in GL_n(\mathbb{R}) : \det(A) = 3\}$ in a way analogous to the way that the single element $1$ of $\mathbb{Z}_2$ can really be thought of as the infinite set $2\mathbb{Z} + 1 = \{\dots,-3,-1,1,3,\dots\}$ of $\mathbb{Z} / 2\mathbb{Z}$, which is no surprise since we are dealing with quotient groups after all. Determinants of matrices behave just like the real numbers with respect to their multiplication.

Another example can be seen in the complex numbers.

Consider the set of nonzero complex numbers, $\mathbb{C}^*$. Again, they form a group with respect to multiplication. The modulus, $\varphi: \mathbb{C}^* \to \mathbb{R}$ such that $a + bi \mapsto |a+bi| := \sqrt{a^2 + b^2}$, is a homomorphism into the reals, since

$$|xy| = |x||y| \text{ for } x, y \in \mathbb{C}$$

as is also easily proven (use the relationship between the modulus and the complex conjugate!)

Since $\ker(\varphi) = \{x \in \mathbb{C}^* : |x| = 1\}$ (the unit circle in the complex plane), the fundamental theorem tells us (with the knowledge that $\varphi$ is surjective into the nonnegative reals) that

$$\mathbb{C}^* / \{x \in \mathbb{C}^* : |x| = 1\} \cong \varphi(\mathbb{C}^*) = \mathbb{R}_{>0}$$

So if we take the complex plane and ‘mod out’ by the unit circle, we end up with the positive real axis! The geometric interpretation of this is that we have identified complex numbers (two-dimensional vectors) whose differences lie in the kernel, that is, whose difference is a unit complex number. Each set of identified vectors only differ in their direction, since multiplying by a unit complex number doesn’t change the magnitude. The plane is partitioned into an infinite family of circular sets of points, one for each possible length of vector (there is one for each nonnegative real number) from the origin, with each set containing all possible angles from $0$ to $2\pi$.

The fact that this quotient gives us the reals is perhaps surprising at first glance, but by considering a well-known homomorphism through the lens of the fundamental theorem it becomes obvious.

To sum up, the power of the fundamental theorem of homomorphisms is that it identifies quotient maps (non-injective homomorphisms) with quotient groups (groups of cosets). They are really two sides of the same coin.

In full generality, it says that **a structure-preserving map achieves exactly the same thing as a quotient object**.

Lastly, another neat feature of the theorem worth mentioning is that it gives us a different vantage point from which to view the notion of a normal subgroup. The set of normal subgroups of a group $G$ is precisely the set of kernels of homomorphisms from $G$ out to some other group. Clearly if we adopt this viewpoint it becomes easier to see why normality is required when forming a quotient group.

buy branded cialis 5 mg cialis free cialis samples Okfaau xyjgyx

Greetings! I’ve been reading your weblog for quite

a long time now and ultimately got the bravery to proceed to supply you with a shout out from Huffman Tx!

Just wished to let you know continue the good work!

my page … SignePKnoy

200 mg viagra

This piece of writing is acgually a good one it hlps new web users, who are wishing in favor oof blogging. Roseanne Kelbee Tychonn

stromectol capsules cialis cheap biaxin capsules Indevw yzzapd

american national insurance

ketoconazole price 2.5 cialis order sumycin online Avvobp lkufta

fildena 100 mg online

generic floxin online pharmacy viagra augmentin online Qofugj eemadz

generic tinidazole https://viagronline.com/ terramycin generic Hjnvli ltmbeh

chloramphenicol capsules https://sildviag.com/ generic ampicillin Emvtwt thzwyz

nationwide auto insurance

roxithromycin for sale order biaxin roxithromycin capsules Cumoyd uajvva

i love this recommended article

keflex tablets http://okbiotic.com/ buy cefadroxil generic Itiknb adneds

canadian pharmacy cialis reviews reputable canadian pharmacy legit canadian pharmacy Xxadzj dqlyrv

best canadian online pharmacy canadian pharmacy mall cialis online pharmacy Fyaqmv ybcrxy

vipps approved canadian online pharmacy https://okpharmp.com/ canadian pet pharmacy Rkkwoz advsxq

fast shipment and great customer service.

this watch is absolutely amazing.

and it’s not an inexpensive item (2-3x most costly than some alternatives.

essay buy online – buy dissertation online essay custom writing

Excellent goods from you, man. I have have in mind your stuff prior to and you are just extremely magnificent. I actually like what you have acquired here, certainly like what you are stating and the way in which through which you say it. You make it entertaining and you still take care of to stay it wise. I can not wait to read far more from you. That is really a tremendous site.

I was very happy to find this internet-site.I wanted to thanks to your time for this glorious read!! I positively enjoying each little little bit of it and I’ve you bookmarked to check out new stuff you weblog post.

It’s perfect a chance to make some plans for the long run and it’s time for you to be at liberty.

I have got learn this post and if I may I desire to counsel you

few interesting things or advice. Maybe you could write next articles associated with this article.

I prefer to read more issues about it!

my blog – ElkeYWyett

I am really enjoying the theme/design of your weblog.

Do you ever run into any web browser compatibility issues?

A couple of my blog visitors have complained about my blog not operating correctly in Explorer but looks great

in Safari. Do you have any recommendations to help fix this problem?

It’s hard to come by well-informed people about this topic, however, you sound like you know what you’re talking about!

Thanks

Hello, I desire to subscribe for this weblog to obtain most recent updates, so

where can i do it please assist.

I will right away clutch your rss feed as I can not to find

your e-mail subscription hyperlink or newsletter service.

Do you’ve any? Please permit me understand so that I may subscribe.

Thanks.

Hello terrific website! Does running a blog like this take a large amount of work?

I’ve absolutely no understanding of coding but I had

been hoping to start my own blog in the near future.

Anyway, if you have any recommendations or techniques for new blog owners please share.

I know this is off topic but I just wanted to ask.

Thank you!

It’s wonderful that you are getting ideas from this post as well as from our argument made at

this place.

Admiring the commitment you put into your site and detailed information you offer.

It’s great to come across a blog every once in a while that isn’t the same outdated rehashed material.

Fantastic read! I’ve bookmarked your site and I’m including your RSS feeds to my Google

account.

I am now not sure where you are getting your information, however good topic.

I needs to spend a while studying much more or working out more.

Thank you for magnificent information I used to be

looking for this information for my mission.

I am regular reader, how are you everybody? This post posted at this site is really nice.

Howdy I am so grateful I found your webpage, I really found you by accident, while I was searching on Yahoo for something else, Anyways I am here now and would just like to say thanks a lot for

a tremendous post and a all round interesting blog (I also

love the theme/design), I don’t have time to read it all at

the minute but I have bookmarked it and also added your RSS feeds, so when I have time I will be back to read a great deal more, Please do keep up the great

work.

Useful info. Fortunate me I found your website by

accident, and I’m stunned why this twist of fate did not came about in advance!

I bookmarked it.

Great blog you have got here.. It’s hard to find good quality writing like yours

these days. I really appreciate individuals like you!

Take care!!

Hi there, just wanted to tell you, I enjoyed this blog post.

It was funny. Keep on posting!

Hi to every one, it’s actually a fastidious for me to go to see this web site, it includes precious Information.

What a material of un-ambiguity and preserveness of precious

know-how concerning unpredicted feelings.

What’s up, this weekend is good in support of me, because this time i am reading this enormous educational paragraph here

at my residence.

Woah! I’m really enjoying the template/theme of this blog. It’s simple, yet effective.

A lot of times it’s very difficult to get that “perfect balance”

between usability and visual appearance. I must say you have done a superb job

with this. Also, the blog loads super fast for me on Opera.

Exceptional Blog!

Simply desire to say your article is as amazing. The clearness in your post is just

excellent and i could assume you’re an expert on this subject.

Well with your permission let me to grab your RSS feed to keep updated with forthcoming post.

Thanks a million and please carry on the enjoyable work.

Ridiculous quest there. What occurred after?

Good luck!

I have learn a few excellent stuff here. Certainly worth bookmarking for revisiting.

I wonder how much effort you set to make one of these fantastic informative website.

Good info. Lucky me I recently found your website by chance (stumbleupon).

I have book-marked it for later!

I am actually thankful to the holder of this website who has shared this enormous piece of writing at here.

Hi! I know this is somewhat off topic but I was wondering

if you knew where I could find a captcha plugin for my comment form?

I’m using the same blog platform as yours and I’m having trouble finding

one? Thanks a lot!

I have read so many content concerning the blogger lovers however this article

is genuinely a pleasant paragraph, keep it up.

For most recent news you have to go to see the web and on world-wide-web I found this web site as a best site for latest updates.

Hey are using WordPress for your blog platform?

I’m new to the blog world but I’m trying to get started and create my

own. Do you require any coding knowledge to make your own blog?

Any help would be greatly appreciated!

Thank you for the auspicious writeup. It in fact was a amusement account it.

Look advanced to more added agreeable from you! By the way, how can we communicate?

Piece of writing writing is also a excitement,

if you be acquainted with then you can write or else it

is complex to write.

Wow, incredible weblog format! How lengthy have you ever been blogging

for? you made running a blog look easy. The whole glance of your site is excellent, let alone

the content!

It’s really a nice and useful piece of info.

I’m happy that you shared this helpful information with us.

Please keep us up to date like this. Thanks for sharing.

Great delivery. Sound arguments. Keep up the great spirit.

I like it when folks get together and share ideas. Great site, keep

it up!

Hello, I check your blogs regularly. Your humoristic style is witty,

keep up the good work!

Hello all, here every person is sharing these familiarity,

therefore it’s nice to read this website, and I used to pay a visit this web site daily.

Hello it’s me, I am also visiting this web page daily, this site is

in fact nice and the users are truly sharing good

thoughts.

I’m extremely inspired together with your writing talents as neatly

as with the layout on your weblog. Is this a paid theme or did you modify it yourself?

Either way keep up the nice high quality writing,

it’s uncommon to look a nice weblog like this one nowadays..

hey there and thank you for your info – I’ve definitely picked

up anything new from right here. I did however expertise some technical

points using this website, as I experienced

to reload the website a lot of times previous to I could get it to

load correctly. I had been wondering if your web hosting is OK?

Not that I am complaining, but slow loading instances times will

sometimes affect your placement in google and could damage your quality score

if advertising and marketing with Adwords.

Anyway I’m adding this RSS to my email and can look out for a lot more of your respective

fascinating content. Ensure that you update this again soon.

Hi to every body, it’s my first pay a visit

of this weblog; this webpage carries amazing and

in fact fine material in favor of readers.

I love your blog.. very nice colors & theme. Did you design this website yourself or did you hire someone to do it for you?

Plz reply as I’m looking to create my own blog and would like to find out where u got this from.

thanks

It’s awesome to go to see this web site and reading the views of all mates on the topic of this post, while I am

also zealous of getting experience.

What’s up to every body, it’s my first visit of this blog; this webpage consists of remarkable and truly good data in favor of visitors.

Thanks for sharing your thoughts on website. Regards

Hello, I log on to your blog on a regular basis.

Your humoristic style is witty, keep doing what you’re doing!

I am actually glad to read this web site posts which includes

tons of valuable information, thanks for providing such statistics.

Undeniably believe that which you stated. Your favorite

justification appeared to be on the net the

easiest thing to be aware of. I say to you, I definitely get irked while people consider

worries that they plainly don’t know about. You managed to hit the

nail upon the top and defined out the whole thing without having side effect , people can take a signal.

Will probably be back to get more. Thanks

I am regular visitor, how are you everybody? This

piece of writing posted at this web page is genuinely nice.

Spot on with this write-up, I honestly believe this site needs a

great deal more attention. I’ll probably be back again to read through more, thanks for the info!

Hi there, just became alert to your blog through Google,

and found that it is really informative. I am going

to watch out for brussels. I will appreciate if you continue this in future.

Many people will be benefited from your writing. Cheers!

My partner and I stumbled over here by a different web address

and thought I might check things out. I like what I

see so now i’m following you. Look forward to exploring your web page repeatedly.

These are genuinely impressive ideas in regarding blogging.

You have touched some fastidious points here. Any way keep up wrinting.

Hurrah, that’s what I was looking for, what a information! present here at this web site, thanks admin of this site.

I am sure this post has touched all the internet users, its

really really nice piece of writing on building up new website.

Stunning quest there. What happened after? Take care!

Generally I don’t read post on blogs, but I would like

to say that this write-up very compelled me to take a look at and

do so! Your writing taste has been amazed me.

Thank you, very nice article.

Hey I know this is off topic but I was wondering if you knew of any widgets I could add to my blog that automatically tweet my

newest twitter updates. I’ve been looking for a plug-in like this for

quite some time and was hoping maybe you would have some experience with something

like this. Please let me know if you run into anything.

I truly enjoy reading your blog and I look forward to your new updates.

Informative article, exactly what I needed.

Magnificent beat ! I would like to apprentice while you

amend your website, how can i subscribe for a blog web site?

The account helped me a acceptable deal. I had been a little bit acquainted

of this your broadcast offered bright clear idea

It’s awesome to visit this website and reading the

views of all mates on the topic of this paragraph, while I am also eager

of getting familiarity.

You really make it seem so easy with your presentation but I find this topic to

be really something that I think I would never understand.

It seems too complex and very broad for

me. I am looking forward for your next post, I will try

to get the hang of it!

I know this site provides quality dependent content and additional information, is there any other site which offers such things in quality?

I’m not that much of a online reader to be honest but your

blogs really nice, keep it up! I’ll go ahead

and bookmark your website to come back later on. Cheers

I always used to study article in news papers but now as I

am a user of internet thus from now I am using net for posts,

thanks to web.

Hello, I think your website might be having browser compatibility

issues. When I look at your blog site in Opera, it looks

fine but when opening in Internet Explorer, it has some overlapping.

I just wanted to give you a quick heads up!

Other then that, great blog!

Hmm it looks like your website ate my first comment (it was super long) so I guess I’ll just sum it up what I had written and say, I’m thoroughly enjoying your blog.

I as well am an aspiring blog blogger but I’m still new to the whole thing.

Do you have any points for newbie blog writers? I’d really appreciate it.

You have made some good points there. I looked on the internet for more information about the issue and found most individuals will go

along with your views on this website.

Hello! I’ve been reading your blog for some time

now and finally got the courage to go ahead and give you a shout out from Kingwood Texas!

Just wanted to mention keep up the great job!

Pretty nice post. I just stumbled upon your weblog and wanted

to say that I have really enjoyed surfing around your blog

posts. In any case I’ll be subscribing to your feed and I hope you write again very soon!

Heya i am for the first time here. I came across this board and I find It

really useful & it helped me out much. I

hope to give something back and aid others like you

helped me.

This is my first time pay a quick visit at here and i am

in fact happy to read everthing at one place.

Have you ever thought about including a little

bit more than just your articles? I mean, what you say is fundamental and everything.

Nevertheless think of if you added some great images or videos to give your posts more, “pop”!

Your content is excellent but with pics and video clips,

this blog could certainly be one of the greatest in its niche.

Awesome blog!

I have been surfing online greater than three hours as of late, but I never discovered any

attention-grabbing article like yours. It’s beautiful price enough for me.

In my opinion, if all web owners and bloggers made excellent content as you did, the web shall be much more useful than ever before.

Pretty element of content. I simply stumbled upon your blog and in accession capital to claim that I get in fact enjoyed account your weblog posts.

Anyway I will be subscribing on your augment and even I achievement you access persistently fast.

Hello there, I found your web site by the use of Google at the same time

as searching for a similar topic, your website came up,

it appears to be like good. I have bookmarked it in my google bookmarks.

Hi there, simply became alert to your blog thru Google, and found

that it is truly informative. I am gonna be careful for brussels.

I will appreciate in case you proceed this in future. A lot of folks shall be benefited

from your writing. Cheers!

I am genuinely delighted to glance at this weblog posts

which includes lots of helpful information, thanks for

providing these data.

This design is incredible! You definitely know how to keep a reader amused.

Between your wit and your videos, I was almost moved to start my own blog (well,

almost…HaHa!) Fantastic job. I really loved what you had to say, and

more than that, how you presented it. Too cool!

This piece of writing is truly a good one it helps new

the web people, who are wishing in favor of blogging.

Hello, i read your blog from time to time and i own a similar one and i was just curious if you get a lot of

spam comments? If so how do you stop it, any plugin or anything you

can suggest? I get so much lately it’s driving me mad so any help is very

much appreciated.

It’s actually a great and helpful piece of information.

I’m glad that you shared this useful information with us.

Please stay us informed like this. Thanks for sharing.

constantly i used to read smaller content which as well clear their

motive, and that is also happening with this paragraph which

I am reading at this place.

I for all time emailed this weblog post page to all my associates, as

if like to read it after that my links will too.

I was curious if you ever thought of changing the page layout of your site?

Its very well written; I love what youve got to say.

But maybe you could a little more in the way of content so people could connect with it better.

Youve got an awful lot of text for only having one or 2 images.

Maybe you could space it out better?

Write more, thats all I have to say. Literally, it seems as

though you relied on the video to make your point.

You clearly know what youre talking about, why throw away

your intelligence on just posting videos to your weblog when you could be giving us something enlightening to read?

Hey I know this is off topic but I was wondering if you knew of any widgets I could add to my blog that automatically tweet my newest

twitter updates. I’ve been looking for a plug-in like this for quite some time and was hoping maybe you would have some

experience with something like this. Please let

me know if you run into anything. I truly enjoy reading your blog and I look forward to your new updates.

Right here is the right web site for everyone who wants to understand this topic.

You realize so much its almost tough to argue with you (not that I actually would want to…HaHa).

You definitely put a new spin on a subject that’s

been written about for years. Excellent stuff, just excellent!

Heya! I’m at work browsing your blog from my new iphone!

Just wanted to say I love reading your blog and look forward to all your posts!

Keep up the superb work!

Hi there i am kavin, its my first occasion to commenting anywhere, when i read this paragraph

i thought i could also create comment due to this brilliant paragraph.

There is noticeably a bundle to know about this. I assume you made certain nice points in features also.

ATT: xanderlewis.co.uk / Xander Lewis – Assorted thoughts. WEBSITE SERVICES

This notification RUNS OUT ON: Sep 19, 2020

We have not gotten a payment from you.

We have actually attempted to contact you yet were not able to contact you.

Kindly Check Out: https://bit.ly/3ktEVeV .

For info and also to process a optional payment for solutions.

09192020194001.