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 ...
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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 ...
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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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Question on the classification of groups of order 102
I have a question regarding the classification of groups of "small" order; we'll take groups of order $102=2 \cdot 3 \cdot 17$ as an example.
Let G be a group of order 102, note that $P_{17}\...
- 149
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3 divides the cardinality of the center of a group of order 225
Let G be group of order $225=3^2 \cdot 5^2$, where the unique (by Sylow's theorems) 5-Sylow subgroup is non cyclic; let's denote this subgroup by $P$. Show that $3$ divides $\left| Z(G) \right|$, ...
- 149
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Group of order 231, show the existence of a normal subgroup of order 21
Let G be a group of order $231=3 \cdot 7 \cdot 11$, by Sylow theorems we know that there exists $P_{3}\in Syl_{3}(G)$ and $P_{7}\in Syl_{7}(G)$ with $P_{7}\trianglelefteq G$.
Clearly by defining $N=P_{...
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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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Sylow $p$-subgroup and Sylow $q$-subgroup both normal
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
$\...
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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 $q$-...
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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 $P_{11}$ ...
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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 ...
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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 $7, 5^2$...
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$G$ 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' \...
user854757
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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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Prove: Group of order $2^{13}\cdot 13$ has a normal subgroup
Prove that a group of order $2^{13}\cdot 13$ has a proper normal subgroup.
Let $n_2$ be the number of sylow $2$ subgroups and $n_{13}$ the number of $13$ sylow subgroups. By the sylow theorems have $...
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Calculating the number of Sylow $5$- and $7$-subgroups in a group of order $105$
I was reading the solution to a problem, but there's a part I really don't understand.
So basically, you have to calculate the $5$-Sylow and the $7$-Sylow subgroups of a group G of order $105$, and by ...
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There are no simple groups of order $480$
Before this writing the smallest order for which the nonexistence of simple groups of that order is not explicitly demonstrated on this site is 480; this self-answered question aims to fill that gap. (...
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There are no simple groups of order $336$
Before this writing the smallest order for which the nonexistence of simple groups of that order is not explicitly demonstrated on this site is $336$; this self-answered question aims to fill that gap....
- 93k
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There are no simple groups of order $264$
Before this writing the smallest order for which the nonexistence of simple groups of that order is not explicitly demonstrated on this site is $264$; this self-answered question aims to fill that gap....
- 93k
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Corollary of the First Theorem of Hall for finite soluble groups
I have to prove the following statement:
If $G$ is a finite soluble group and $N \trianglelefteq G$, then any $\pi$-Hall subgroup of $G/N$ is of the type $HN/N$ for some $H$ a $\pi$-Hall subgroup of $...
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Prove any two nonabelian groups of order 4301 are isomorphic
I'm tasked with proving (for homework) any two nonabelian groups of order $4301 = (11)(17)(23)$ are isomorphic. I've proved that for any nonabelian group $G$, we have $G = P_{11}N$, with $P_{11} \cap ...
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Unique $p$-Sylow subgroup of Dihedral group $D_n$.
If $p>2$ is a prime number that divides $n \in \mathbb N$, then $D_n = \langle r,s \rangle$ (with $r^n=s^2=id, sr=r^{-1}s$) has a unique $p$-Sylow subgroup $H \leq D_n$.
I'm having trouble with ...
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If $H\leq G$ and $G$ has a Sylow $p$-subgroup, then so does $H$
Consider the following question:
Let $H$ and $P$ be subgroups of a finite group $G$. Show that the sets
$HxP$, $x \in G$, partition $G$. By considering the action of $H$ on
the set of left cosets of $...
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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 ...
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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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a doubt on Sylow's Theorem in Dummit&Foote's Abstract Algebra
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 ...
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When does $|G|=pq$ admit unique non abelian representation? [duplicate]
Classifying $G$ with $|G|=pq$ has been asked several times in this site, but I am having trouble with number theory. Let $q<p$, $Q=\langle u\rangle\in \text{Syl}_q(G)$, $P=\langle v\rangle\in \text{...
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$N \cong (\mathbb{Z}/p\mathbb{Z}) \rtimes_{\varphi} (\mathbb{Z}/p\mathbb{Z})^{\times}$ where $N$ is the normalizer of Sylow $p$-subgroup of $S_p$
Question: Suppose $p$ is prime, $P \subset S_p$ is a Sylow $p$-subgroup of $S_p$ (the symmetric group on $p$ elements) and $N = N(P)$ is the normalizer of $P.$ Show that
$$ N \cong (\mathbb{Z}/p\...
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Do normalizers of $p$-subgroups of $S_n$ always grow?
Recently I encountered, the 'normalizers grow principle' which gives an equivalent characterization of Nilpotent groups.
Looking at the other 'extreme' case of $S_n$, my question is:
Let $n \geq 4$. ...
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Sylow P-Subgroups of Normal Subgroups
Just as a note, please don't give me the proof, I feel like I'm close and am just looking for a point in the right direction.
I am trying to prove that if $G$ is a finite group with $|G|$ = $p^nm$, ...
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Dummit & Foote's Abstract Algebra: Error on proof of Sylow's theorem (Theorem 4.5.18)
I think I have spotted an error in the proof of Theorem 4.5.18 (line 139) on Dummit & Foote's Abstract Algebra. Or maybe I am missing something?
On the proof of $ r \equiv 1 \pmod {p} $, it is ...
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Example Sylow Subgroup of $A_5$
I have been given the problem of determing $n_2(A_5)$.
I do not want the answer to this, since I am making some progress by myself. The issue is, I have run into what seems like a contradiction, and I ...
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$3$-Sylows of $A_7$
The problem states: "Find the number and the structure of the Sylow $3$-subgroups of $A_7$."
Things I know:
-No problem with the structure, it is $C_3 \times C_3$ as they're no elements of ...
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How many copies of $D_4$ in $S_5$?
This question is motivated by studying the Sylow-$p$-subgroups of $S_5$.
From Sylow's theorems, the order of Sylow-2-subgroups has order 8 and are all conjugate to copies of the dihedral groups of $...
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Showing that a group of order $72$ has a normal subgroup of order at least $3$
From Section $2.12$ of Herstein's "Topics in Algebra" ($2^{\text{nd}}$ edition):
$\;$ We give one other illustration of the use of the various parts of Sylow's theorem. Let $G$ be a group ...
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To construct a group with certain properties.
Let $G$ be a finite group with properties listed in the following.
(1) $G/G''\cong A_4$.
(2) $G/K\cong SL(2,3)$ where $K=F(G)$ is isomorphic to an extraspecial $3$-group of order $3^7$, $K/Z(K)$ is a $...
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For odd prime $p$, $\Bbb F^\times_{p^2}\subset\mathrm{GL}_2(\Bbb F_p)$
$\newcommand{\sl}{\mathrm{SL}}\newcommand{\gl}{\mathrm{GL}}$Let $G = \sl_2(\Bbb F_p)$ with $p$ and odd prime. Prove that the Sylow $l$-subgroup of $G$ are cyclic for every odd prime $l$ dividing $|G|$....
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Why does $K=\Bbb F_p[\theta_n]$ contains subfield of order $p^e$ with solutions to $X^{pe}-X=0$ over $K$ where $e$ is the exponent of $p\pmod n$?
Why does $K = \mathbb{F}_p [\theta_n]$ contains subfield of order $p^e$ with solutions to $X^{pe}-X = 0$ over $K$ where $e$ is the exponent of $p\pmod n$?
The setting is as follows. We have $p$ prime,...
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Help understanding proof: classifying groups of order $21$
This proof has something I just didn't understand, apologies if the question is too basic. I'll focus on the part I don't understand (after some background).
Let $|G|=21$. So $G$ has a unique Sylow 7-...
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Determination of the Groups of Order $99$
Would the following brief argument suffice in showing that there are (up to isomorphism) only two groups of order $99$, being $Z_{99}$ or $Z_{3} \oplus Z_{33}$? I have seen rather "fuller" ...
- 145
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2
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Prove every Sylow Subgroup of $G$ is Cyclic
I'm working on the following problem. Let $G$ be a finite group such that for each $n \mid |G|$, we have that $G$ contains at most $1$ subgroup of order $n$. Prove that every Sylow $p$ subgroup of $...
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Group Homology of a Sylow $p$-Subgroup
I am trying to understand the proof of the following proposition:
Proposition Let $G$ be a finite group and let $P$ be a normal Sylow $p$-subgroup of $G$. Then for all $n$ there is an isomorphism
$$
...
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