# 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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### Why is a Sylow 5-subgroup abelian?

For weeks I tried to solve the following question on Brilliant: Fill in the blank: "Every group of order ___ is abelian." And these are the possible answers I get: 15, 16, 20, 21, 27. Using ...
1 vote
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### "Simple" group of order $1004913$ problem, fixed point part

Let $G$ be a group of order $1004913 = 3^3 \cdot 7 \cdot 13 \cdot 409$. We suppose that $G$ is simple. We want to obtain a contradiction. This is the Exercise 29 in Chapter 6.2 of Dummit-Foote. As ...
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### Group of quaternions $Q_8$ is not a subgroup of $S_4$

I'm having a hard time conceptualising Sylow p-subgroups and i'm looking for help with the following problem. I'm more so looking for understanding the Sylow p-subgroups than the other steps of the ...
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### Some questions about the proof of the First Sylow theorem

I try to understand the proof of the First Sylow theorem: A finite group whose order is divisible by a prime $p$ contains a Slow $p-$subgroup. The proof is that Let $S$ be the set of all subsets of ...
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### Characteristic subgroups of a group of order 6 or 12

I am trying to understand the proof of the statement that if $|G| = 60$ and $G$ has more than one Sylow $5$-subgroup, then $G$ is simple. This statement and its proof can found in Page 145 of Dummit ...
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### 2-Sylow subgroup of $S_6$. [closed]

I'm asked to find the 2-Sylow subgroups of the symmetric group $S_3$ using a certain method that is to find the 2-sylow subgroup of $S_{2^k}$. This method is illustrated in I.N.Herstein's "TOPICS ...
1 vote
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### Theorem Reference Query: $p'$-Elements in Supersolvable Groups

I have the following question: In the context of a finite supersolvable group $G$, where $p$ represents the largest prime divisor of $|G|$, I would like to inquire about a specific theorem, for which ...
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### Show that a group of order $2010$ with an abelian normal subgroup of order $6$ is abelian

Let $G$ be group of order $2010$ with $N$ an abelian normal subgroup of order $6$; show: $N\le Z(G)$ where $Z(G)$ is the center of $G$ Show that there exist a unique $5$-Sylow subgroup of $G$ ...
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### Let $H,K \leq G$ s.t $H,K$ two different Sylow $p$-subgroups of $G$. Prove $HK$ isn't a subgroup of $G$.

First of all I wish to explain why I'm opening a new thread: I want to know if my proof is correct. The only proof I could find relies on the assumption $H,K$ are distinct which isn't always the case:...
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Let $G$ be a finite group and $p\neq q$ prime numbers that divide $\|G\|$. Let $P$ and $Q$ be a Sylow $p$‐subgroup and a Sylow $q$‐subgroup of $G$, respectively. If $n_p=1$ and $n_q=1$, show that $\... 2 votes 1 answer 87 views ### Number of Sylow subgroups in$p$-solvable groups - Navarro article This is a article which Gabriel Navarro wrote. I'm reading lemma 2.1. I see that "By standard arguments, recall that in any coprime action, if$q$is a prime, then every$A$-invariant$q$-... 0 votes 1 answer 168 views ### Show that in a group$G$of order$165=3\cdot 5\cdot 11$, the center$Z(G)$contains a subgroup of order 11. I have solved most of the problem, but I still can't figure out the last part. If$P_{11}\in Syl_{11}(G)$, then$n_{11}\equiv 1 \pmod{11}$and$n_{11} \mid 3\cdot 5$, i.e.$n_{11}=1$, so that$P_{11}$... 1 vote 2 answers 56 views ### A group may have two distinct characteristic Sylow-$p$-subgroups There is a statement: A group may have two distinct characteristic Sylow-$p$-subgroups. I think it is not necessarily true because if there is unique Sylow-$p$-subgroups, it is characteristic; if ... 10 votes 1 answer 182 views ### Group of order$1575$is solvable Let$G$be a group of order$1575 = 3^25^27$. Show that$G$is solvable. So far, my idea is to show that$n_7, n_5 $or$n_3$equals$1$, so that there exists a unique Sylow subgroup of order$7, 5^2$... 4 votes 2 answers 100 views ###$G$finite metabelian group,$P$a Sylow$p$-subgroup. Show that$P'$is abelian and normal in$G$Let$G$be a finite metabelian group and let$P$be a Sylow$p$-subgroup of$G$. I have to observe that the derived subgroup$P'$is abelian and normal in$G$. Since$G'$is abelian, the subgroup$P' \... 106 views

### In what sense does Sylow theory determine "local structure" of a group?

I'm reviewing some group theory with Alperin and Bell's Groups and Representations, and they chose to title the chapter containing the Sylow theorems, $p$-groups and compositions series by Local ...
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### Classification of groups such that the converse to Lagrange holds [duplicate]

I was remeditating Sylow subgroups recently, after reading somewhere that it served as a partial converse to Lagrange's theorem. After a bit more pondering I started wondering if we can find the ...
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### Let $G$ be a group of order $30$. Show that if $G$ is nonabelian, there are more than one $2$-subgroups of Sylow.

Let $G$ be a group of order $30$. Show that if $G$ is nonabelian, there are more than one $2$-subgroups of Sylow. Suppose there is only one $2$-subgroup of Sylow and show that $G$ is abelian. It is ...
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### If $|G|=2^2\cdot 5 \cdot 19\Rightarrow n_5=n_{19}=1$

By using the Sylow Theorems we need $n_5\in \{1,2^2\cdot 19\}$ and $n_{19}\in\{1,20\}$. First Case: If $n_5=2^2\cdot 19$ and $n_{19}=20$, the group cannot stand the pressure: we have too many elements....
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### Sylow subgroup of $G$ with order the smallest prime factor of $|G|$

Problem: Let $p$ be the smallest prime dividing the order of the finite group $G$. If $P \in Syl_p(G)$ and $P$ is cyclic prove that $N_G(P) = C_G(P)$. I know the standard proof constructing an ...
1 vote
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### if $G$ is a finite nilpotent group $n$-generated, then is any Sylow subgroup at most $n$-generated?

If $G$ is a pronilpotent group, then $G$ is the product of its Sylow subgroups (like in the finite case). Suppose that $G$ is finitely generated, then can be a Sylow subgroups infinitely generated? ...
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I have a doubt in Dummit&Foote's Abstract Algebra on page142 : Corollary 20. Let $P$ be a Sylow $p$-subgroup of $G$ . Then the following are equivalent: (1) $P$ is the unique Sylow $p$-subgroup ...