Smallest variety containing $\mathbb{Z}$

I've been told that the answer is all abelian groups, but I don't see how.

I know that the class of all nilpotent groups of degree 1 is a group variety and that a group being nilpotent of degree 1 is equivalent to that group being abelian so the class of all abeliani groups is indeed a variety. I also know that the integers are a group when considered with addition as the operation and that a class of algebraic structures of the same signature is a variety if and only if it is closed under the taking of homomorphic images, subalgebras and (direct) products, but I'm not quite sure how to use this here or if this is even the way to go about this.

I think that the main structure of this proof should be to show that $\mathbb{Z}$ is contained inside the variety of all abelian groups and then show that its the smallest one that could possibly contain $\mathbb{Z}$. I know that in the class of abelian groups, the language is $(+, 0, \frac{1}{})$ and the identities are $$(x+y)+z = x + (y+z),$$ $$x+y = y+x,$$ $$x + 0=x,$$ $$x+(-x) = 0,$$ all of which obviously hold on $\mathbb{Z}$. Is that enough to say that the class of abelian groups contains $\mathbb{Z}$?

How would I show nothing smaller could contain $\mathbb{Z}$?

• You can use the easy part of the HSP theorem : any variety is stable under homomorphisms, products, and subvarieties. An immediate corollary is stability under coproducts. May 18 '18 at 6:04
• @Max Thank you, but I really don't understand how. I haven't seen any examples and I can't seem to find any tutorials online about this subject matter. Could you please elaborate more on how exactly? I have a proposition in my notes that says that for a family of algebras in a variety the direct product is in the variety May 18 '18 at 6:14
• @ericwofsey I didn't know that was an option for a tag. Very useful information. Thank you May 18 '18 at 6:17
• What part don't you understand ? The HSP theorem or coproducts ? (I'll try to write a detailed answer) May 18 '18 at 6:24
• @Max That's very kind of you. Honestly, I don't understand any of it. I definitely don't know how to apply the HSP thoerem. Is my argument that the class of all abelian groups is a variety because its equivalent to the variety of all nilpotent groups of degree 1. Does this allow us to conclude that $\mathbb{Z}$ is contained in such a variety? What are we trying to show is closed under all those things? Is it $\mathbb{Z}$ or the variety of all abelian groups May 18 '18 at 6:33

I'll give an answer that does not depend on the HSP theorem.

Let $\mathsf{V}(\mathbb{Z})$ be the smallest variety containing the group $\mathbb{Z}$. Let $\mathcal{A}$ be the variety of all abelian groups. It should be clear that $\mathsf{V}(\mathbb{Z})\subseteq\mathcal{A}$ since $\mathbb{Z}$ is an abelian group. To show the other inclusion, we assume by way of contradiction that $\mathbb{Z}$ satisfies an equation that is not satisfied by all abelian groups. Let's call that equation $s\approx t$.

1. If $s\approx t$ is satisfied, then so is $s-t\approx 0$.

2. The equation $s-t\approx 0$ is of the form $m_1x_1+\dots+m_kx_k\approx 0$ for some integers $m_1,\dots,m_k$.

3. If $m_1x_1+\dots+m_kx_k\approx 0$ is satisfied, then so are the equations $m_1x\approx0$, $m_2x\approx0$, $\dots$, $m_kx\approx 0$.

4. If the equations $m_1x_1\approx0$, $\dots$, $m_kx\approx 0$ are satisfied, then so is $nx\approx0$ where $n=\gcd(m_1,\dots,m_k)$.

But $\mathbb{Z}$ does not satisfy $nx\approx0$ for any integer $n$; a contradiction!

The details still need to be filled in. If you can't figure them out yourself, then I recommend consulting Cliff Bergman's ''Universal algebra: fundamentals and selected topics.''

• What exactly does $\approx$ mean? Why isn't it just equal? May 18 '18 at 21:41
• I use $\approx$ for equations of terms built up from variables. For example: $x\cdot(y\cdot z)\approx(x\cdot y)\cdot z$. I use $=$ after I have plugged in elements of a specific algebra. For example: $a\cdot(b\cdot c)=(a\cdot b)\cdot c$. It's more a habit of mine rather than a terribly important distinction.
– Eran
May 19 '18 at 17:10
• thanks. How did you get from $m_1x_1 + ... m_kx_k \approx 0$ to each term $m_ix_i \approx 0$? May 19 '18 at 18:56
• $m_ix \approx 0$* May 19 '18 at 19:01
• I still can't figure it out. I've even asked it as a separate question. Could you please explain? May 22 '18 at 2:57

It seems you use the definition if variety given by " a class of algebras defined by a certain set of equations".

In that case, one can prove the HSP theorem (which some authors use as a definition of variety) : let $V$ be a variety containing $\mathbb{Z}$.

This variety is stable under products (that's the P in HSP) so it contains all the $\mathbb{Z}^X$, $X$ a set.

But now $\mathbb{Z}^{(X)}$ (the sequences of $\mathbb{Z}^X$ that have all coordinates $0$ except perhaps for a finite number of them) is a subalgebra of $\mathbb{Z}^X$ so is in $V$ (that's the S in HSP).

That's now the interesting part : if $G$ is an abelian group, then $G$ is a homomorphic image of $\mathbb{Z}^{(G)}$ (that's a standard exercise in group theory), hence $G$ is in $V$ (that's the H in HSP).

This shows that all abelian groups are in $V$ But you also proved that all algebras in $V$ were abelian groups, so $V$ is precisely the variety of abelian groups !

Note that I've used the fact that the free abelian groups (the $\mathbb{Z}^{(X)}$ for $X$ a set) were in $V$, which came from the fact that the free abelian group on one generator ($\mathbb{Z}$) was in $V$. As soon as you can show that a variety contains the free algebras of $V$, then you can conclude in exactly the same way I did.