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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_ …

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 …

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 …

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 …

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 …

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 …

Since $G$ is a finite abelian group we have $\widehat{G}\cong G$. For $\chi\in\widehat{G}$ to be Galois-stable, or just stable under complex conjugation, its order must be at most 2. However $\widehat …

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 …

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 …

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 …

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) …

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 …

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 …

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 …

There are (at least) two explicit formulas. First, $D(n)$ can be shown to be a multiplicative function (this means $\gcd(m,n)=1$ implies $D(mn)=D(m)D(n)$), and the values of $D(p^k)$ for all primes $p …