# 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.

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### Topological generator of profinite group

I'm trying to prove that $\pi(\operatorname{Spec}\mathbb{Z}[\frac{1}{2}])$ is topologically generated by its $2$-Sylows (Exercise 6.30 of Lenstra's notes on Galois Theory for Schemes). I've been given ...
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### Question on uniqueness of minimal normal subgroup

Let $H$ be minimal normal $p$-sylow subgroup of a group $G$. Order of $G$ is $ap^{m}$ where gcd$(a,p^m) = 1$ Since $H$ is normal $p-SSG$ it implies that is unique of that order i.e. $H$ is the only ...
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### Proving a group of order $35^3$ is solvable

Is my thinking correct when asked to show that a group $G$ of order $35^3$ is solvable, I first show that by the sylow theorems there exists a sylow $p$-subgroup of order $5^3$ and another unique ...
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### If abelian $P\in{\rm Syl}_p(G)$, and $H\le P$ and $H^g\le P$, show $g\in G$ is the left product of an element of $N_G(P)$ with an element of $C_G(H)$

If abelian $P \in{\rm Syl}_p(G)$, and $H \leq P$ and $H^g \leq P$, show $g \in G$ is the left product of an element of $N_G(P)$ with an element of $C_G(H)$ First, I set $g=nc$, where I hope to find ...
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### Prove that if a finite group $G$ is soluble and has more than one $17$-Sylow subgroup, then it has more than $100$ $17$-Sylow subgroups.

Prove that if a finite group $G$ is soluble and has more than one $17$-Sylow subgroup, then it has more than $100$ $17$-Sylow subgroups. My main reason for asking: Do people find this easy or hard? ...
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### Difficulty using Sylow theorems

I am really struggling to apply Sylow's theorems to various problems, so any help is much appreciated. For part (a) here, since we are supposing that $H<G$, I tried to use an element $g$ of $G$ ...
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### Trouble using Sylow's theorems

So, I found part (a) fairly easy, and also managed part (c) assuming (b) to be true. Any help with (d) and especially (b) would really be appreciated. My issue with part (b) is understanding which of ...
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### Every finite group $G$ such that $p\mid |G|$ has a maximal p-subgroup

I'm reading the book "An Introduction to the Theory of Groups" by Joseph Rotman. Maximal $p$-subgroup is defined as: $P$ is a maximal $p$-subgroup of a group $G$ if for every $Q\le G$, $Q$ ...
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### How many equivalence classes are there in $J$.? [closed]

Let $G$ be the symmetric group $S_5$ of permutations of five symbols. Consider the set $J$ of subgroups of $G$ that are isomorphic to the non-cyclic group of order $4$. Let us call two subgroups $H$ ...
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### Does a group of order 24 have a sylow 11 subgroup

I am going through Algebra In Action by Shahriari and on page 147 it says Definition 7.8 (Sylow subgroups). Let G be a finite group with |G| = $p^a$m, where $p \nmid m$, and $a$ is a non-negative ...
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### Notation for a proof containing group actions

I am writing a proof to Sylow's 2nd Theorem. I have done it using group actions. My tutor told me it would be wise to introduce notation to make it more clear what group action I am talking about. I ...
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### A group of order $150$ has at least $4$ conjugacy classes made by elements of order a power of $5$?

Let be $G$ a group of order $150$ and $g \in G\,\,$ s.t. $|g|=25\,$: Prove that $\exists \,\,K\lhd G\,\,$s.t. $K \simeq C_5$. $\,\,$ Say whether $\exists \,\,h \in G\,\,$ s.t. $|h| =15$.$\,\,$ Prove ...
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### A problem about Sylow Theorem [closed]

I'm learning Sylow Theorem. My textbook uses the following result directly and does not tell me why. I don't know how to prove it. $C_{p^rn}^{p^r}\equiv n \mod pn$. Here $p$ is a prime number and $r,n$...
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### $|G| = pqr^2$, $G$ not simple [closed]

If $G$ is a group with cardinality $pqr^2$ and $p,q,r$ different primes $\ge 2$, and given $1+np$ doesn't divide $qr^2$ for any $n \in \mathbb{Z}^+$, How can you show $G$ is not simple? I feel like ...
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### $2$-Sylow of a subgroup of $A_n$ [closed]

I don't know how to start to prove this: Let $H$ be a subgroup of $S_n$ and let $T$ be a subgroup of $H$ such that $T$ is a $2$-Sylow of $H$. Show that $H$ is conntained in $A_n\iff$ $T\le A_n$. Maybe ...
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### Which group is simple? [duplicate]

I am currently looking whether a group G of the following order is simple or not $ord(G)=100$ $ord(G')=200$ We have first that for $G$, $ord(G)=2^25^2$. For $G'$ we have that $ord(G')=2^35^2$. Now ...
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### Application of Sylow to a group of order 60

Let $G$ be a group with $|G|=60=2^2 \cdot 3 \cdot 5$ and assume $G$ has a unique $3$-Sylow subgroup $H_3$. The claim is that then $G$ has also a unique $5$-Sylow subgroup. This is the procedure to ...
### Conjugation action on a semi-direct product $N\rtimes H$
Let $G$ be a finite group, with $H\le G$ and $N\trianglelefteq G$ such that $G=N\rtimes H$. Suppose that the conjugation action of $H$ on $N$ induces $2$ orbits in $N$. Prove the following requests: ...