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The group algebra $k[G]$ of a finite group $G$ over a field $k$ knows little about $G$ most of the time; if $k$ has characteristic prime to $|G|$ and contains every $|G|^{th}$ root of unity, then $k[G]$ is a direct sum of matrix algebras, one for each irreducible representation of $G$ over $k$, so $k[G]$ only knows the dimensions of the irreducible representations of $G$.

The group ring $\mathbb{Z}[G]$, on the other hand, knows at least as much as $k[G]$ for every $k$ (which one can obtain by tensoring with $k$); this information should let you deduce something about the entries in the character table of $G$, although I'm not sure exactly what. If the characteristic of $k$ divides $|G|$ then $k[G]$ knows something about the modular representation theory of $G$, although again, I'm not sure exactly what. So:

How strong of an invariant is $\mathbb{Z}[G]$? More precisely, what group-theoretic properties of $G$ do I know if I know $\mathbb{Z}[G]$? What are examples of finite groups $G_1, G_2$ which are not isomorphic but which satisfy $\mathbb{Z}[G_1] \cong \mathbb{Z}[G_2]$?

If in addition to $\mathbb{Z}[G]$ we are given the augmentation homomorphism $\mathbb{Z}[G] \to \mathbb{Z}$ then it looks like we can recover the group cohomology and homology of $G$, so it seems plausible that $\mathbb{Z}[G]$ contains quite a lot of information.

At a minimum, setting $k = \mathbb{R}$ I think we can compute the Frobenius-Schur indicator of every complex irreducible representation of $G$.

Edit: It seems the second problem is known as the isomorphism problem for integral group rings and is quite hard.

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Doesn't the edit already answer the question? This is a well-known area of group theory with lots of literature. Here is a survey: ubbcluj.ro/books-online/matematica/Marcus_Markus/cluj5.pdf –  Martin Brandenburg Apr 22 '13 at 18:28
    
@Martin: well, it answers the second question but not (I think?) the first. I would be happy to accept that survey as an answer if you posted it. –  Qiaochu Yuan Apr 22 '13 at 18:57

2 Answers 2

If you know that $G$ and $H$ are at least abelian, then $\mathbb{Z}[G]\cong\mathbb{Z}[H]$ implies $G\cong H$.

proof: The homology groups of $G$ and $H$ are independent of the choice of resolution up to canonical isomorphism, and the groups are defined by $H_iG=H_i(F_G)$ where $F$ is a projective resolution of $\mathbb{Z}$ over $\mathbb{Z}G$ (similary for $H$). Since $\mathbb{Z}G\cong\mathbb{Z}F$, the $G$-modules $F_i$ can be regarded as $H$-modules via restriction of scalars and hence the projective resolution $F$ for the homology of $G$ can also be used for the homology of $H$. We have $F_G\cong F_H$ since $\mathbb{Z}\otimes_{\mathbb{Z}G}F\cong \mathbb{Z}\otimes_{\mathbb{Z}H}F$ by the obvious map $1\otimes f\mapsto 1\otimes f$ [using the isomorphism $\varphi:\mathbb{Z}G\rightarrow \mathbb{Z}H$ we have $1\otimes f=1\otimes gf\mapsto 1\otimes \varphi(g)f=1\otimes f$], and so $H_i(G)\cong H_i(H)$ for all $i$. In particular, $G/[G,G]\cong H_1(G)\cong H_1(H)\cong H/[H,H]$. Since $G$ and $H$ are abelian groups, $[G,G]=0=[H,H]$ and hence $G\cong H$.

If we knew the augmentation map then we can recover $G_{ab}$ as $I/I^2$, where $I$ is the augmentation ideal (kernel of the augmentation map). But even if we don't know the augmentation map, $\mathbb{Z}[G]\cong\mathbb{Z}[H]$ would imply $I_G\cong I_H$ and hence $G_{ab}\cong H_{ab}$, reproducing my above statement.

And in general there does exist counterexamples, I forgot the sources on them, but this MO question refers to one of them:
http://mathoverflow.net/questions/60609/strong-group-ring-isomorphisms

(I first claimed, without thinking, that $G$ sits as the group of units in $\mathbb{Z}[G]$, which is what Qiaochu refers to in his comment below).

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This doesn't work. You need to know the augmentation map to know either the augmentation ideal or the module structure of $\mathbb{Z}$. –  Qiaochu Yuan Apr 22 '13 at 18:23
    
What doesn't work?, my proof was answering your third question. I never claimed we knew the augmentation map. –  Chris Gerig Apr 22 '13 at 21:12
    
You start talking about the homology groups of $G$ and $H$, but these can't be defined from $\mathbb{Z}[G]$ and $\mathbb{Z}[H]$ alone. You need to know the augmentation map. –  Qiaochu Yuan Apr 22 '13 at 21:19
    
Again, this (my post) has nothing to do with your first two questions -- you asked about the implication $\mathbb{Z}[G]\cong\mathbb{Z}[H]\Rightarrow G\cong H$, to which I show you get $G_{ab}\cong H_{ab}$. –  Chris Gerig Apr 23 '13 at 4:07
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Obviously the augmentation map exists, but there's no reason to believe that an isomorphism between two group rings will respect augmentation maps. I think you're confused here. –  Qiaochu Yuan Apr 24 '13 at 23:11

You might consult Chapter 9 in Milies, Sehgal, "An Introduction to Group Rings". It discusses how several invariants of groups are encoded in their integral group rings, and how this leads to an affirmative answer of the integral group ring isomorphism problem for some classes of groups.

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