# Tagged Questions

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### 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 ...
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### 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?
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### 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 ...
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### 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.
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### 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 ...
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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 $N=\sum_{g\in ... 0answers 63 views ### Finite$\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$. Then ... 1answer 68 views ### Eigenspace decomposition for semisimple module Let's start with a prime$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$. I want to find a decomposition into eigenspaces ... 2answers 181 views ### Why is this statement about generators of groups true? Let$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 $$G/H = ... 1answer 52 views ### proving that a action of hopf algebra k(G) on A implies a G-grading on A Let k(G) be the hopf algebra of functions on G with values in k with pointwise multiplication and a comultiplication given by \Delta(f)(x,y) = f(xy) and let A be a k-algebra. I read (in ... 1answer 168 views ### Chinese remainder theorem and isomorphism Suppose that there are different primes p_1 and p_2 and the group C_{p_1}. I would like to decompose the following tensorproduct and thought of using the chinese remainder theorem to get ... 1answer 82 views ### G/G' \cong I/I^2 where I is the augmentation ideal [duplicate] Possible Duplicate: Isomorphism between I_G/I_G^2 and G/G' Let G be a finite group. Let I\unlhd\mathbb{Z}[G] be the augmentation ideal of the integral group ring \mathbb{Z}[G]. ... 1answer 68 views ### The indecomposable projective \mathbb{F}_pG-module with UJ/J\cong \mathbb{F}_p Let: G be a finite group; p be prime; J be the Jacobson radical of \mathbb{F}_pG. A paper I'm trying to read mentions the following object: The indecomposable projective ... 1answer 148 views ### Smallest pure subgroup containing a fixed subgroup I will ask a slightly more precise question then in the title. Let G be a finite abelian group, and g_1, \ldots, g_n \in G such that the cyclic groups they generate are in direct sum \langle g_1 ... 3answers 234 views ### Splitting exact sequences of finite abelian groups I would like to find a condition for an exact sequence of abelian groups$$ 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= ...
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 ...
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 ...