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It is not hard to prove $|\mathrm{Out}(T)|\leqslant \log_p|T|$ when $T=L(q)$ is a simple group of Lie type of characteristic $p$. (One uses formulas for $|\mathrm{Out}(T)|=dfg$ and for $|T|$ for diffe …

I claim that the conjecture is false, and there is a counterexample with $G=S_5$, $n=2$, and $A$ cyclic of order 6. In this case $\gamma_2(G)=[G,G]=G'=A_5$ has order $5!/2=60$. Take $A$ to be the abel …

Imprecise question: To get a normalizing field outer automorphism of order $r$, must we multiply the dimension by $r$? Precise hypothesis: Let $p\geqslant 5$ be a prime, let $q$ be a power of $p$ and …

The answer to this question is No. Let $U$ be the natural module for $H=\textrm{SL}(2,5^5)$ and let $V=U\otimes U^\sigma\otimes\cdots\otimes U^{\sigma^4}$ where $\sigma$ is field automorphism $\lambda …

The answer to your both questions is 6. Consider the symmetric group $S_3=\langle a,b\mid a^2=b^3=1, b^a=b^{-1}\rangle$, and take $P_1=\langle a\rangle$, $P_2=\langle ab\rangle$, $P_3=\langle ab^2\ran …

Let me speak to part of your question: What after it? We can not reasonably expect to classify the $p^{2n^3/27 +O(n^{8/3})}$ groups of order $p^n$ for large $n$. We can classify $p$-groups with partic …

You seem to be saying that a finite $p$-group which has a non-normal subgroup must have a cyclic center. A counterexample when $p=2$ is the group $G=D_8\times C_2$ of order 16. Its center is $C_2\time …

See the following paper: A. Lucchini, F. Menegazzo, M. Morigi. On the existence of a complement for a finite simple group in its automorphism group. Special issue in honor of Reinhold Baer (1902–1979) …

One uses a matrix version of Hilbert's Satz 90 to answer Geoff's comment "If the field $E$ is not this minimal field, it seems less obvious to me how to realise the representation over the subfield ge …

The comment by Frieder Ladisch suggests to me that considering exponents may be relevant. Suppose that we generalize Stefan Kohl's function $f_p(n)$ as follows: Definition: Fix a prime $p$ and an exp …

I take it Pablo your question can be rephrased as follows. Does there exist an epimorphism $\tau\colon A\ltimes C\to A$ where $A$ acts irreducibly on $C$ and where $\ker(\tau)\ne C$? If this is your q …

This is an interesting question, even though it is not well defined. Call a group "good" if it has a "good" bijection between its conjugacy classes and its irreducible complex representations. I agree …

Your first problem has a simple solution. Suppose $p$ is a prime and $(n!)_p$ is the $p$-part of $n!$. Dirichlet proved $(n!)_p=p^k$ where $k = (n-s_p(n))/(p-1)$ and $s_p(n)$ is the sum of the base-$p …

First let me paraphrase the question. Given integers $m,n,k$ each at least 2, set $d:=\max(m,n,k)+2$. Do there exist elements $a,b$ in the symmetric group $S_d$ such that $|a|=m$, $|b|=n$ and $|ab|=k$ …