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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 …
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 …
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: …
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 …
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 …
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 …
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:
…
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 …
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 $G \cong H$.
…
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 …
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 …
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/ …