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Questions about the branch of algebra that deals with groups.

6 votes

Group rings over central products

I don't have a reference, but the proof is routine so that the result should be well-known. Also note that $H, K$ don't need be finite. For a proof first note that if $H, K$ are subgroups of $G$ and …
Todd Leason 's user avatar
4 votes
0 answers
167 views

For $P$ $\mathbb{Z}G$-projective, $\mathbb{Q}\otimes P$ is $\mathbb{Q}G$-free

I'm looking for a proof of a theorem of Swan [1, Theorem 3]: If $G$ is a finite group and $P$ a finitely generated projective $\mathbb{Z}G$-module, then $\mathbb{Q}\otimes_\mathbb{Z}P$ is a free …
Todd Leason 's user avatar
8 votes
Accepted

Groups with trivial rational homology and their finite index subgroups

Thompson's group $T$ gives an example, i.e. if $T \le H$ has finite index, then $H^\ast(H;\mathbb{Q}) \neq 0$. More specifically, there is always a non-trivial class in $H^4(H;\mathbb{Q})$. Proof: …
Todd Leason 's user avatar
5 votes
1 answer
209 views

Minimal number of algebra generators of a group ring

Let $k$ be a commutative ring with unit and let $(A,\varepsilon)$ be a (not necessarily commutative) augmented, finitely generated $k$-algebra with augmentation ideal $I$. If $\mu(A)$ denotes the mi …
Todd Leason 's user avatar
4 votes

A table for irreducible integral representation of finite cyclic groups

For cyclic groups $C_p$ of prime order $p$ the irreducible integral representations are known (I don't know if there are results for cyclic groups of composite order but it's likely since the result f …
Todd Leason 's user avatar
6 votes
0 answers
344 views

Transfer-free proof that the power map is a homomorphism

For a central subgroup $C$ of finite index of a group $G$ it is well-known that the power map $$G \to C,\;g \mapsto g^{(G:C)}$$ is a group homomorphism. This is commonly proved by help of the transfe …
Todd Leason 's user avatar
13 votes
6 answers
2k views

Sylow theorems for infinite groups

Are there classes of infinite groups that admit Sylow subgroups and where the Sylow theorems are valid? More precisely, I'm looking for classes of groups $\mathcal{C}$ with the following properties: …
Todd Leason 's user avatar
2 votes

Linear Independence & Group Theory

Linear algebra can be applied to exploit the structure of extraspecial groups. That's a p-group $G$ where the commutator subgroup equals the center $C$ and has order p. $V:= G/C$ is an elementary abel …
Todd Leason 's user avatar
4 votes
1 answer
178 views

Applications of the Roggenkamp-Scott theorem ?

In 1987 Roggenkamp and Scott published a solution of the integral isomorphism problem for $p$-groups, i.e. if $G,H$ are $p$-groups and $\mathbb{Z}[G] \cong \mathbb{Z}[H]$ as rings then $G \cong H$. …
Todd Leason 's user avatar
6 votes
0 answers
135 views

Groups with decomposable automorphism group

I wanted to ask what is known about finite groups whose automorphism group is decomposable (i.e. $Aut(G)=H \times K$ for some groups $H,K$) ? Is there for instance some kind of classification or do s …
Todd Leason 's user avatar
7 votes
1 answer
186 views

Restrictions for presenting groups with cyclic quotient

Let $H$ be a group, $\phi$ an automorphism of $H$ of order n and fix $h_0 \in H$. I wonder, what the restrictions are, such that $$G:= \lt H,g \mid g^n=h_0,\quad \forall h \in H: ghg^{-1}=\phi(h) \gt …
Todd Leason 's user avatar
9 votes
1 answer
1k views

Counting isomorphism classes via extensions

Given a group $Q$ and an abelian group $C$, I want to determine the number $I(Q,C)$ of isomorphism classes of all groups $G$ having a central subgroup $C'$ isomorphic to $C$ such that the quotient $G/ …
Todd Leason 's user avatar