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Does anyone know a reference to the following results, which I can prove, but I suspect may be known. Let $R(n)$ denote the number of primitive $n$th roots of unity with positive real part, and $L(n)$ …

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 …

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 …

@Marty Isaacs: There exist non-nilpotent groups whose conjugacy class sizes are all squares. For example, let $G$ be Magma's 93rd group order 540. It has class sizes 1,4,9. Indeed, |G|=15*1+30*4+45*9. …

I have a question regarding a partial order $<$ on the set ${\rm Part}(n)$ of partitions of $n$. Given $\lambda=(\lambda_1,\lambda_2,\ldots)\in{\rm Part}(n)$ with $\sum_{i\geq1} \lambda_i=n$ and $\lam …

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 …

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) …

Let $p$ be a prime and let $F$ be a free group of rank $d\geq 1$. Kostrikin [1] proved that the $d$-generated Burnside group $B=B(d,p)=F/F^p$ of exponent $p$ has a maximal finite quotient $\overline{B …

A group $G$ is called a $k$-orbit group if its automorphism group ${\rm Aut}(G)$ acting naturally on $G$ has precisely $k$ orbits. The only finite 2-orbit groups are the elementary abelian groups $(C_ …

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 …

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 …

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 …

In the case that $n$ is prime and $H$ is normal in $G$. Your second questions has an affirmative answer. The composition length is $1$, $n$, or $1+d$ where $d\mid (n-1)$, in each case at most $n$. See …

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 …