Mikko Korhonen
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 Sep 6 awarded Good Question Apr 14 awarded Necromancer Nov 5 comment Lucido's three prime lemma Also, it is possible that a nonsolvable group has the property in question. For example $\operatorname{PSL}(3,3)$ has order $2^4 \cdot 3^3 \cdot 13$ and contains an element of order $6$. Sep 24 awarded Autobiographer Jun 9 awarded Self-Learner May 12 comment Applications of Frobenius theorem and conjecture There is also a nice direct proof by Brauer (in the paper "On A Theorem of Frobenius" in AMM), which does not require induction at all. Basically if $n = p^\alpha m$, $p$ prime, $(m,p) = 1$, then using a subgroup of order $p^\alpha$, we can define an equivalence relation $\sim$ on $G$ such that each $\sim$-class has order $p^\alpha$, and the set of solutions to $x^n = 1$ is an union of $\sim$-classes. This proof also gives you more general statements about the number of solutions in a fixed double coset $HyH$, for example. May 7 awarded Yearling Feb 2 comment Index of agemo subgroups in $p$-groups @GerryMyerson: Yes, the name comes from the common notation $\mho_1(G) = G^p$. Jul 3 awarded Popular Question Mar 10 awarded Teacher Feb 16 answered Applications of Frobenius theorem and conjecture Feb 14 awarded Enthusiast Feb 5 answered Lower bounds on the number of elements in Sylow subgroups Jan 23 revised Lower bounds on the number of elements in Sylow subgroups added 4 characters in body Jan 23 awarded Editor Jan 23 revised Lower bounds on the number of elements in Sylow subgroups deleted 22 characters in body; added 129 characters in body Jan 23 awarded Nice Question Jan 23 comment Lower bounds on the number of elements in Sylow subgroups @Nick: Ok, I'll edit the post. What I mean by the construction is this. Fix a prime $p$ and integer $n \geq 1$. The construction shows that we can find a group $G$ with Sylow $p$-subgroups of order $p^n$ (not just some arbitrary power of $p$ like in your comment) such that $f_p(G)$ is arbitrarily large. Jan 21 comment Lower bounds on the number of elements in Sylow subgroups Perhaps one way to start with this is that given $n$, $p$ and $k$, find the smallest possible value for $f_p(G)$. I don't know if the bound $2p^{n+1}−p^n$ is sharp for the case $k=2$. It is sharp for all primes $p$ such that $2p+1$ is prime, which can be seen by the construction in my question. Jan 21 awarded Scholar