Maschke's Theorem and split extensions

Question

Recall that a short exact sequence of groups like $1\longrightarrow A\longrightarrow E\longrightarrow G\longrightarrow 1$ is called an extension of $A$ by $G$. Now, let $k$ be a finite field and $G$ a finite group. Let $A$ be a $k[G]$-module. Does every extension of $A$ by $G$ is split if, and only if, the characteristic of $k$ does not divide the order of $G$. I know that is true for short exact sequences of $k[G]$-modules by Maschke's Theorem but I don't know how to use this theorem for my question.

Now , suppose that G is a p-group and $ k$ a finite field with $car(k)/|G|$. Does exist a k[G]-module A admiting a central Frattini extension?.

Can anybody help me, please? I would appreciate any hints and comments. Thank you in advance!

Answer 1

The ``if''' part is true by the Schur-Zassenhaus theorem (actually the Schur part suffices).

The ``only-if'' part is false in general -- take e.e. $G=S_3$ and $A=C_2^2$ the reduced permutation module.

What is true is a slightly weaker version that if the characteristic of $k$ divides the order of $G$ there exists a module $A$ (indeed -- thanks @Derek Holt -- a simple module) over $k$ such that the extension of $A$ by $G$ is nonsplit. This follows from a theorem by Gaschütz (Über modulare Darstellungen endlicher Gruppen, die von freien Gruppen induziert werden, Math. Z. 60 (1954) 274–286.).

The Schur-Zassenhaus theorem is in any standard book on group theory. The second fact is strangely enough not mentioned in textbooks, though it is a natural question, and it deserves more dissemination.

Answer 2

A technical problem prevents me to putting this as a comment, sorry for this. @ahulpke Thank you very much for your answer. The Schur's theorem states that if $A$ is of order coprime to the order of $G$, then the above extension splits. However, I didn't assume that $|A|\wedge|G|=1$, so why schur's theorem works in this situation. In fact, I was thinking to use $H^{2}(G,A)=Ext_{k[G]}^{2}(k,A)$. @Derek Holt @ahulpke Thank you for pointing out the second fact. If $car(k)/|G|$, it is not clear for me that Gaschütz's theorem implies the existence of an irreducible module $A$ over $k$ such that the extension of $A$ by $G$ is nonsplit. Maybe, I don't have the right version of this theorem. Can you please putting it here or giving a hint. Thank you again. @ahulpke