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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 …

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 …

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: …

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 …

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 …

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 …

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: …

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 …

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 $G \cong H$. …

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 …

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 …

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/ …