# Tagged Questions

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

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### There are 267 not isomorphic groups of order 64. How many of these are Abelian? Describe them. [on hold]

There are 267 not isomorphic groups of order 64. How many of these are Abelian? Describe them.
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### Group order 168 having a normal subgroup of order 4, then G has a normal subgroup of order 28.

Prove that if G is a group of order 168 that has a normal subgroup of order 4, then G has a normal subgroup of order 28. resolution: P4 is the ordeme subgroup 4 and P7 the order subgroup 7. As P4 is ...
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### Need help in understanding a certain step of a certain proof in finite group theory and group actions

A proof is from Aluffi's textbook "Algebra: Chapter 0". A statement: There are no simple groups of order $24$. The proof from the book: Let $G$ be a group or order $24 = 2^33$, and consider ...
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### On the maximum number of Sylow subgroups

I have some doubts regarding the first sentences below the fourth case in the more elementary solution by Prof. Samuel to problem 11856 of the Monthly. Here you have a screenshot of the relevant part ...
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### Sylow subgroup related

In a group of all invertible $4 \times 4$ matrices with enteries in the field of $3$ elements, What is the cardinality of any $3$- sylow subgroup?
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### Let $G$ be a group and $N$ be a minimal normal subgroup of $G$, $G_p \in Syl_p(G)$, where $p$ is an odd prime divisor of $|G|$.

Let $G$ be a group and $N$ be a minimal normal subgroup of $G$, $G_p \in Syl_p(G)$, where $p$ is an odd prime divisor of $|G|$. Suppose that $M/N$ be a maximal subgroup of $G_pN/N$. then $M = PN$ for ...
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### Sylow $p$-subgroup of a finite group

I know: Let $P$ be a Sylow $p$-subgroup of a finite group $G$.If $N$ is normal in $G$, then $P \cap N$ is a Sylow $p$-subgrup of $N$. But if $N$ is not normal in $G$ , there is also the issue? ...
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### Sylow p-subgroups of a finite group G which are contained in some other subgroup H are also Sylow p-subgroups in H

I have found a proof to the following question but I don't quite understand it. Let $P \leq H \leq G$ and $|G|=p^\alpha m$, where $p$ doesn't divide $m$. I need to show that if $P \in Syl_p(G)$, ...
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### Representation of a finite group and its Sylow $p$-subgroup

Let $G$ be a finite group with order $|G|=p^n \cdot m$ for some positive integers $n,m$ and $H$ be a Sylow $p$-subgroup of $G$. What relations can we say about the representations of $G$ and $H$?
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### There is no simple group of order $1452$.

I want to prove that there is no simple group of order $1452$. We have $1452 = 2^2\cdot 3\cdot 11^2$, and the Sylow theorems give: \begin{align} n_2 &\in \{1,3,11,33,121,363\} \\ n_3 &\in \{...
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### G is cyclic if it is a finite p-group and has only one maximal subgroup [duplicate]

How can I show that for a finite $p$-group $G$,$G$ is cyclic if it has just one non trivial maximal subgroup?
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### Find some subgroups of a group with 36 elements

Let a group $G$ with $\left | G \right | = 36$, then $G$ contains a normal subgroup of order $9$, $18$ or $3$. I can see groups of order $9$ in the case $n_{3} = 1$, by Silow's Theorem, but I do not ...
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### $G$ contains a normal $p$-Sylow subgroup and $p$ divides the order of the center

I am looking at the following: Let $G$ be a non-abelian finite group with center $|Z|>1$. I want to show that if $G/Z$ is a $p$-group, for some prime $p$, then $G$ contains a normal $p$-Sylow ...
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### Analogue of Third Sylow Theorem for sets

Let $n=p^em$ such that $p^e$ is the largest power of $p$ that divides $n$, and $p\nmid m$. Let $N$ be the number of subsets of order $p^e$ in a set $S$ with $|S|=n$. I want to compute $N\mod p$. Does ...
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### How to prove a group of order $144$ is not simple using **Normalizers of Sylow intersections**.

How to prove a group of order $144$ is not simple using Normalizers of Sylow intersections. Here's what I have tried, but I am unable to proceed further. And if I proceed with Sylow-$2$ subgroups ...
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### Show that if $\sigma=(a_1,a_2,\dots,a_m)$ and $\tau$ is any element of $S_n$, then $\tau\sigma\tau^{-1}=(\tau a_1,\tau a_2,\dots,\tau a_m)$ [duplicate]

Show that if $\sigma=(a_1,a_2,\dots,a_m)$ and $\tau$ is any element of $S_n$, then $\tau\sigma\tau^{-1}=(\tau a_1,\tau a_2,\dots,\tau a_m)$. I'm not quite sure how to start this. The solution starts ...
Let $G$ be a finite group. Let $K$ be a subgroup of $G$ and let $N$ be a normal subgroup of $G$. Let $P$ be a Sylow $p$-subgroup of $K$. Is $PN/N$ is a Sylow $p$-subgroup of $KN/N$? Here is what I ...
### Proof of a group of order $60$ is simple containing sylow $5$ subgroups.
So I was reading Sylow theorems from Dummit and Foote while I came across the following result I've understood the whole proof except the highlighted portion in red circle. Why $|H|\geq 1+4\times 6$? ...