0
votes
1answer
17 views

Minimal Frattini Extensions

From "Construction of Finite Groups" article : "To construct minimal Frattini extensions of $H$ we have to consider all suitable irreducible $H$-modules $M$. Clearly, $M$ can be viewed as an ...
1
vote
1answer
38 views

Extension of a group by a module

What is the definition of an extension of a group by a module? and what is the difference between extensions by groups and modules?
1
vote
1answer
37 views

Property of G-modules involving the invariant elements under the G-action

I am stuck at some basic fact I would like to prove. I tried proving it using $G-$orbits and cardinalities, but without success. Let $p$ be some prime number, $G$ be a finite $p-$group and $A$ a ...
6
votes
1answer
73 views

The cardinality of the group $\mathrm{SL}(k,\mathbb{Z}/n\mathbb{Z})$

I want to calculate the cardinality of the group $\mathrm{SL}(k,\mathbb{Z}/n\mathbb{Z})$, $k,n \in \mathbb{N}$. When $n$ is prime, it is easy to calculate, so I want to know the general case.
1
vote
0answers
69 views

Lift a group action from a quotient

Let $p$ be a rational prime and $H$ be a finite cyclic group of prime order $l$ prime to $p$, i.e. $(l,p) = 1$. Let $G$ be a finite abelian group of $p$-power order. If I can write an (abelian) group ...
1
vote
2answers
87 views

The norm map in group cohomology is an isomorphism if $M$ is a projective $G$-module

This is exercise III$.1.1(c)$ in Brown's "Cohomology of Groups." Let $G$ be a finite group, $M$ a $G$-module, and $\overline{N}:M_G\to M^G$ the norm map defined by $[m]\to Nm$, where $N=\sum_{g\in ...
1
vote
0answers
62 views

Finite $\Delta$-module of $p$-power order

I have a question concerning lemma, that I want to prove: Let $p$ be a prime and $\Delta$ be a finite group of order prime to $p$. Let $M$ be a finite $\Delta$-module of order a power of $p$. Then ...
1
vote
1answer
66 views

Eigenspace decomposition for semisimple module

Let's start with a prime $p$ and a group $\Delta$ of order prime to $p$. Let $M$ be a finite $\mathbb{F}_p[\Delta]$-module of order a power of $p$. I want to find a decomposition into eigenspaces ...
6
votes
2answers
181 views

Why is this statement about generators of groups true?

Let $G$ be free abelian of rank $n$ and $H \subseteq G$ a subgroup also of rank $n$. It is known that $G/H$ is finite, in fact a direct sum of at most $n$ cyclic groups. Thus we can write $$G/H = ...
1
vote
1answer
52 views

proving that a action of hopf algebra k(G) on A implies a G-grading on A

Let $k(G)$ be the hopf algebra of functions on $G$ with values in $k$ with pointwise multiplication and a comultiplication given by $\Delta(f)(x,y) = f(xy)$ and let $A$ be a $k$-algebra. I read (in ...
4
votes
1answer
168 views

Chinese remainder theorem and isomorphism

Suppose that there are different primes $p_1$ and $p_2$ and the group $C_{p_1}$. I would like to decompose the following tensorproduct and thought of using the chinese remainder theorem to get ...
0
votes
1answer
82 views

$G/G' \cong I/I^2$ where $I$ is the augmentation ideal [duplicate]

Possible Duplicate: Isomorphism between $I_G/I_G^2$ and $G/G'$ Let $G$ be a finite group. Let $I\unlhd\mathbb{Z}[G]$ be the augmentation ideal of the integral group ring $\mathbb{Z}[G]$. ...
2
votes
1answer
68 views

The indecomposable projective $\mathbb{F}_pG$-module with $UJ/J\cong \mathbb{F}_p$

Let: $G$ be a finite group; $p$ be prime; $J$ be the Jacobson radical of $\mathbb{F}_pG$. A paper I'm trying to read mentions the following object: The indecomposable projective ...
4
votes
1answer
148 views

Smallest pure subgroup containing a fixed subgroup

I will ask a slightly more precise question then in the title. Let $G$ be a finite abelian group, and $g_1, \ldots, g_n \in G$ such that the cyclic groups they generate are in direct sum $\langle g_1 ...
3
votes
3answers
232 views

Splitting exact sequences of finite abelian groups

I would like to find a condition for an exact sequence of abelian groups $$ 0\to H\to G\to K\to 0 $$ to split. Assume for simplicity that $H=\langle h \rangle$ is cyclic, and choose a basis for $G= ...
3
votes
3answers
158 views

Constructing a basis for finite abelian groups

Let $G$ be a finite abelian group, and $g_1, \ldots, g_k$ a set of "linearly independent elements", namely such that $\langle g_1 \rangle \oplus \ldots \oplus\langle g_k \rangle$. I would like to ...
3
votes
1answer
153 views

Linear algebra of finite abelian groups

Let $\phi:G \to H$ be a surjective homomorphism of finite abelian groups, and let $g_1, \ldots, g_n$ be an irredundant set of generators (from now on, a basis) for $G$. be a basis for $G$, meaning a ...