# Tag Info

### Name of algebraic structure (group-like)

This is a commutative quasigroup. However, personally I don't find this sort of naming to be particularly illuminating. To my mind the simplest way to think about this operation $\star$ is actually as ...
• 368k
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### (Why) Is there no analogue to the classification of finitely generated abelian groups for abelian groups?

No, it is badly false and the classification of abelian groups is much more complicated. For example $\mathbb{Q}$ is not a nontrivial direct sum in any way. Exercise 1: Prove that if an abelian group ...
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### Name of algebraic structure (group-like)

The structure you are describing is called a Steiner quasigroup. The phrase Steiner quasigroup is sometimes abbreviated 'squag', so some folks call these objects squags. In general, a (finite) Steiner ...
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### Motivation in considering abelian groups in modules

Let $G$ be a group, $R$ a ring and assume there is an "$R$-module structure on $G$" (something satisfying the usual axioms except we don't require $G$ to be abelian). Then, let $x,y\in G$ be ...
• 7,930
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### Morphism of free groups that induces isomorphism on abelianizations

The alternating group $A_5$ is generated by $a = (12345)$ and $b = (12)(34)$. These give a transitive action of the free group $F_2$ on $\{ 1, 2, 3, 4, 5 \}$. The stabilizer of $5$ is a subgroup $H$ ...
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### Let $G = \Bbb Z \times \Bbb Z$ with group law given by addition. Let $H$ be the subgroup generated by $(2,3)$. To which group is $G/H$ isomorphic to?

As we've noted, the kernel of your map is strictly larger than your $H$: your $H$ consists only of those elements in which the first entry is a multiple of $2$, and the second entry is the same ...
• 361k
Accepted

### For any $n$, let $\alpha(n)=$ #non-isomorphic Abelian groups of order $n$. What is the maximum value of $\alpha(n)$ for $n\le 200$?

Finite abelian groups are direct sums of their Sylow $p$-subgroups, which are direct sums of cyclic $p$-groups $C_{p^k}$, and this direct sum decomposition uniquely identifies the isomorphism class. ...
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