# Questions tagged [sylow-theory]

For questions about Sylow theorems in the context of group theory. Not for use with questions regarding Sylow systems, which belong in solvable-groups.

986 questions
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### $|G|=p^2q^2$ where p,q are primes, $p<q$ and $p \nshortmid (q^2-1)$ Show $G$ is Abelian.

$|G|=p^2q^2$ where p,q are primes, $p<q$ and $p \nshortmid (q^2-1)$ Show $G$ is Abelian. ($p$ does not divide $q^2-1$) The first thing to note is that $(q^2-1)=(q-1)(q+1)$ so we can conclude that ...
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### How to prove $Q_8$ is not subgroup of $S_4$ by sylow thm?

Prove $Q_8$ is not subgroup of $S_4$ Hi. It is trivial that $D_4$ is a subgroup of $S_4$. But the case $Q_8$ make me confuse why this group is not a subgroup of the $S_4$. Why is the $Q_8$ isn't ...
0answers
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### I have two confusion in sylow theorem formula . True/false

I have two confusion in sylow theorem formula $1.$Order of a $n$ sylow subgroup in $GL_n (\mathbb{F_p})= \frac{\prod_ {i=0}^{n-1} (p^n -p^i)}{(p-1)^n p^{\frac {n(n-1)}{2}}}$ $2.$ Order of a $q-$...
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### Show that any finite nilpotent group of square free order is cyclic.

Show that any finite nilpotent group of square free order is cyclic. Hint: Suppose G is such a group. Any Sylow subgroup of G is of prime order. Hint: Any finite nilpotent group is the direct ...
1answer
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### How can $SN_G(D)=G$ if $S$ is not normal in $G$?

Is there any theorem which says anything like $SN_G(D)=G$ where $S$ is a $p$-Sylow subgroup and $D$ is the intersection it has with some other subgroup? I know the first that will probably spring to ...
0answers
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### trying to understand this proof of sylows theorem that say the number of p-sylow subgroups is 1+kp

I'm Very confused as to what my lecturer means in the final few lines of a proof of one of sylows theorems means. The theorem in question is the one that says the number of sylow P-subgroups is 1+kp. ...
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### a few questions about what's going on in this proof of sylow's theorem I found

Note: If someone wants to even just answer my first question in the comments until someone else decides to give a full answer I'd be pretty happy. I just want to know there's no mistakes in it before ...
1answer
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### In a group of order $m p^n$ for $p$ prime, if $k<n$, is there an element of order $p^k$? [duplicate]

Let $G$ a group of order $mp^n$ where $p$ is prime. Let $k\leq n$. Is there an element of order $p^k$ ? Since $p$ divide $|G|$, by Cauchy theorem, there is $g\in G$ s.t. $g$ has order $p$. I can't do ...