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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.

Sylow's theorems give information about the numbers of subgroups of fixed order a finite group contains.

Often used with and .

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A question about Sylow subgroups and $C_G(x)$

Let $G=PQ$ where $P$ and $Q$ are $p$- and $q$-Sylow subgroups of $G$ respectively. In addition, suppose that $P\unlhd G$, $Q\ntrianglelefteq G$, $C_G(P)=Z(G)$ and $C_G(Q)\neq Z(G)$, where $Z(G)$ is the center of $G$. I want to prove there exist two…
Adeleh
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Is there a geometric idea behind Sylow's theorems?

I have a confession to make: none of the proofs of Sylow's theorems I saw clicked with me. My first abstract algebra courses were more on the algebraic side (without mention of group actions and geometric motivation for groups, except hastily…
ante.ceperic
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Show that in any group of order $23 \cdot 24$, the $23$-Sylow subgroup is normal.

Show that in any group of order $23 \cdot 24$, the $23$-Sylow subgroup is normal. Let $P_k$ denote the $k$-Sylow subgroup and let $n_3$ denote the number of conjugates of $P_k$. $n_2 \equiv 1 \mod 2$ and $n_2 | 69 \implies n_2= 1, 3, 23,…
user58289
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Distinct Sylow $p$-subgroups intersect only at the identity, which somehow follows from Lagrange's Theorem. Why?

It seems that often in using counting arguments to show that a group of a given order cannot be simple, it is shown that the group must have at least $n_p(p^n-1)$ elements, where $n_p$ is the number of Sylow $p$-subgroups. It is explained that the…
Jonathan Beardsley
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How big must the union of a group's Sylow p-subgroups be?

For various orders $n$ it's a common exercise to prove that a finite group $G$ of order $n$ can't be simple by using the Sylow theorems to show that there is some prime $p \mid n$ such that the number $n_p$ of Sylow $p$-subgroups equals $1$, so the…
Qiaochu Yuan
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If H is a subgroup of G, then H has no more Sylow subgroups than G

If $H$ is a subgroup of the finite group $G$, then how do I show that $n_p(H) \leq n_p(G)$? Here $n_p(X)$ is the number of Sylow $p$-subgroups in the finite group $X$. Here is my attempt: Suppose the order of $G$ is $n$, the order of $H$ is $m$,…
user132177
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Maximum number of Sylow subgroups

I've been studying Sylow-$p$ subgroups, and I've come across this problem. Let $G$ be a finite group. Show that the number of Sylow subgroups of $G$ is at most $\frac{2}{3}|G|$ . ($|G|$ is the number of elements of $G$). I am having trouble…
Jephte
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Let G be a nonabelian group of order $p^3$, where $p$ is a prime number. Prove that the center of $G$ is of order $p$.

Let G be a nonabelian group of order $p^3$, where $p$ is a prime number. Prove that the center of $G$ is of order $p$. Proof Since $G$ is not abelian, the order of its center cannot be $p^3$. Since it is a $p$-group, the center cannot be trivial.…
user58289
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Putnam 2007 A5: Finite group $n$ elements order $p$, prove either $n=0$ or $p$ divides $n+1$

Putnam 2007 Question A5: "Suppose that a finite group has exactly $n$ elements of order $p$, where $p$ is a prime. Prove that either $n=0$ or $p$ divides $n+1$." I split this problem into two cases: where $p$ divides $|G|=m$, and where $p$ does not…
Daniele1234
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A Group Having a Cyclic Sylow 2-Subgroup Has a Normal Subgroup.

I want to prove the following: Let $G$ be a group of order $2^nm$, where $m$ is odd, having a cyclic Sylow $2$-subgroup. Then $G$ has a normal subgroup of order $m$. ATTEMPT: We will show that $G$ has a subgroup of order $m$. Let…
caffeinemachine
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Can we construct a group with exactly $k$ Sylow-Subgroups?

Inspired by the answers given by these three questions (here, here, and here), what is the general solution for constructing a group with a specific number of Sylow subgroups? That is, given a prime $p$ and a positive integer $n\equiv1\pmod p$, is…
J. Linne
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Proofs of Sylow theorems

It seems that there are many ways to prove the Sylow theorems. I'd like to see a collection of them. Please write down or share links to any you know.
Karolis Juodelė
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What do Sylow 2-subgroups of finite simple groups look like?

What do Sylow 2-subgroups of finite simple groups look like? It'd be nice to have explanations of the Sylow 2-subgroups of finite simple groups. There are many aspects to the question, so I envision many answers. I tried to give an answer of "broad…
Jack Schmidt
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Group of order $p^{n}$ has normal subgroups of order $p^{k}$

Q: Prove that a subgroup of order $p^n$ has a normal subgroup of order $p^{k}$ for all $0\leq k \leq n$. Attempt at a proof: We proceed by Induction. This is obviously true for $n=1, 2$. Suppose it is true for $m \leq n-1$. Now, take the group $G$…
TheManWhoNeverSleeps
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Classification of groups of order 30

How do I find all the groups of order 30? That is I need to find all the groups with cardinality 30. I know Sylow theorems.
Shiva Prakash
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