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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2
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1answer
35 views

Conjugates of Sylow $p$-groups in $GL_3(F_p)$

In this list of review questions, there is the following question about $GL_3(F_p)$. Question 1.38. Let $G$ be the group of invertible 3 × 3 matrices over $F_p$, for $p$ prime. What does basic ...
2
votes
1answer
21 views

Identifying semidirect products-groups of order 28

So, I'm very new to working with semi-direct products. I'm working on my algebra qual prep, and one of the questions was to identify all the groups of order 28. I'm pretty sure I have the ...
2
votes
0answers
56 views

Does group $G$ of order 42 have a normal cyclic subgroup of order 21?

Show that a group $G$ of order 42 has a normal cyclic subgroup of order 21. What I did so far is using Sylow's theorem to show that $G$ has a unique 7-sylow subgroup $S(7)$ (which is normal) and {1 ...
5
votes
1answer
49 views

Show that there's no simple group of order $63$, please check my reasoning

I want to show that there's no group of order $63$ which is simple and would like to know if my simple reasoning is correct. I am irritated because this is an exercise which is supposed to be harder. ...
4
votes
1answer
41 views

Struggle on Sylow p-subgroup

I'm in trouble on Dummit and Foote Abstract Algebra Ex. 6.2 13: Let $P,Q$ be distinct Sylow p-group with maximal $|P \cap Q|$. Show $N_G(P \cap Q)$ cotains more than one Sylow p-group and each pair ...
7
votes
1answer
108 views

How to show that the group is abelian?

I have this exercise: a. Let $\sigma \in S_{15}$ be an element of order 5. What type of cycles can occur in the decomposition of $\sigma$ in disjoint cycles? b. Let $S \subseteq S_{15}$ be ...
1
vote
1answer
38 views

Abstract Algebra: Proof involving a p-Sylow, nomal subgroup P in G. [closed]

Let $P$ be a normal subgroup in $G$ where $P$ is a $p$-Sylow subgroup of $G$. Show that $\phi(P)=P$ for every automorphism $\phi$ of $G$.
5
votes
2answers
57 views

Must a non-simple group have a normal Sylow subgroup?

In class, one way we're taught to prove a group is not simple is to exhibit a normal Sylow subgroup. I'm wondering if the converse is true, i.e. if a group is not simple, must it have a normal Sylow ...
14
votes
4answers
416 views

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 ...
2
votes
2answers
66 views

Direct product and Sylow subgroups

Let $G$ be a finite group that is equal to inner direct product of its subgroup $P$ and $Q$, where $P$ is a Sylow $p$-subgroup and $Q$ is a Sylow $q$-subgroup of $G$. If $L \le G$, prove that $L$ is ...
4
votes
2answers
80 views

Group of order $135$ abelian and not cyclic

I am trying to solve the following: Let $G$ be a group of order $135$. Show that if $G$ has more than one normal subgroup of order $3$, then $G$ is abelian and non-cyclic. What I could do was: ...
1
vote
2answers
66 views

$|G|=p_1p_2p_3$ distinct primes with $p_i \nmid p_j-1$ then $G$ is cyclic

Problem Let $p_1,p_2,p_3$ be three distinct primes with $p_i \nmid p_j-1$ for all $1\leq i,j \leq 3$ and let $G$ be a group of order $p_1p_2p_3$. Show that $G$ is cyclic. I've tried to come up with ...
4
votes
3answers
57 views

Number of Sylow $p$-subgroups of a direct product of groups

Let $G$ be the group $S_4\times S_3$ . Prove or disprove the following: a $2-$Sylow subgroup of G is normal a $3-$Sylow subgroup of G is normal I've got $|S_4\times S_3|=144$ and the group as not ...
3
votes
1answer
102 views

Sylow p-subgroups and set X not divisible by p

Let $P$ be a Sylow $p$-subgroup of $G$ and suppose that $P\subseteq Z(G)$. Show that the set $X$ of elements of $G$ with order not divisible by $p$ is a subgroup of $G$ and that $G=P\times X$. I ...
3
votes
1answer
70 views

Difference between definitions of $p$-subgroup and Sylow $p$-subgroup

I'm reading Abstract algebra by Dummit and Foote and the following definitions are made: $1$. A group of order $p^{\alpha}$ for some $\alpha\geq1$ is called a $p$-group. Subgroups of $G$ which are ...
3
votes
2answers
63 views

Construct Group of Order 21 Without Semi Product

We have two possibilities, I know that one of the possibilities is the cyclic group$\frac{\Bbb{Z}}{21\Bbb{Z}}$. The other possibility as shown below with Sylow's theorems is $\Bbb{Z}_7 \times ...
-2
votes
1answer
50 views

Sylow's theorem and uniqunes of normal supgroup

Let $G$ be a finite group of order $pq,$ where $p$ and $q$ are primes such that $p < q.$ Then how to prove that $G$ has a unique normal subgroup of order $q?$
0
votes
2answers
30 views

Sylow counting - classifying groups of order 15

let $G$ be a group of order $15$. this is the argument I was given: We have $15 = 3\times 5$ so we start with $p = 5$ We have by Sylow's theorem that $N_5 = 1 \mod 5$ so $N_5 = 1$ or $N_5 \geq 6$. ...
3
votes
1answer
61 views

Various Intersections of Sylow p-subgroups.

I was told yesterday that in a system of Sylow $p$-subgroups of a finite group $G$, if, $\{S_1,S_2, \cdots, S_n\}$ make up the system, it can happen that, say, the intersection of $S_1$ and $S_2$ has ...
1
vote
1answer
77 views

$G$ is a finite group,with eight Sylow 7 subgroups. Show that there exists a normal subgroup $N$ of $G$ s.t. index $[G:N]$ divisible by 56, not by 49.

Let $G$ be a finite group which has exactly eight Sylow 7 subgroups. Show that there exists a normal subgroup $N$ of $G$ such that the index $[G:N]$ is divisible by 56 but not by 49. I will start by ...
0
votes
0answers
27 views

Sylow p,q,r-Subgroups

I am quite new to group theory, so I am trying to get my head around Sylow's Theorems and other stuff....I got an exercice here and I am not sure how to go on with the proofs. We have a group G of ...
-1
votes
3answers
52 views

Proof of normal subgroup

The question is: There's a group G, with order pm, where p is a prime number and mcd(p,m) = 1. We suppose that G has an unique p-Sylow subgroup P. Proof that P is a normal subgroup of G. How I ...
1
vote
3answers
136 views

What is a group action, and how can we apply it to Sylow theory

I am studying Sylow theorems at the moment, more specifically trying to solve the following problem that I recently posted: Let G be a finite group which has exactly eight Sylow 7 subgroups. Show ...
1
vote
1answer
43 views

Why does group action by conjugation on sylow subgroups define a homomorphism into the symmetric group?

Sylow theorems state that sylow p subgroups of a group G are conjugate. Often I see argumentation that if there are n sylow p subgroups in G then we can define a group action on it by conjugation and ...
2
votes
1answer
127 views

If we have exactly 1 eight Sylow 7 subgroups, Show that there exits a normal subgroup $N$ of $G$ s.t. the index $[G:N]$ is divisible by 56 but not 49.

Let $G$ be a finite group which has exactly eight Sylow 7 subgroups. Show that there exits a normal subgroup $N$ of $G$ such that the index $[G:N]$ is divisible by 56 but not by 49. Now this is my ...
0
votes
1answer
122 views

Let $G$ be a group. Let $G/K$ be an abelian group. Prove that $C=\{e\}$.

I'm stuck at this exercise: Let $G$ be a group, with $|G|=pqr$, $p,q,r$ different primes, $q<r$, $r \not\equiv 1$ (mod $q$), $qr<p$. Also suppose that $p \not\equiv 1$ (mod $r$), $p ...
1
vote
1answer
23 views

Sylow $p$-subgroup of a normal group.

Let $G$ be a transitive subgroup of $S_p$ and let $H$ be a non-trivial normal subgroup of $G$. I need to show that any Sylow p-subgroup of $G$ is also contained in $H$. I know that any transitive ...
1
vote
2answers
27 views

Sylow $p$-group of a transitive subgroup of $S_p$

How do I show that any transitive subgroup of $S_p$ contains a non-trivial Sylow $p$ subgroup, of cardinality $p$? I am trying to prove a result of Galois and the only hint I have is that if $p$ is a ...
3
votes
1answer
72 views

The number of Sylow subgroups on $G$ with $|G|=pqr$

I'm doing a part of an exercise and I don't know how to go on. Here it goes: Let $G$ be a group, with $|G|=pqr$, $p,q,r$ different primes, $q<r$, $r \not\equiv 1$ (mod $q$), $qr<p$. Show $G$ ...
2
votes
1answer
50 views

Let $G$ be a group, with $|G|=pqr$, $p,q,r$ different primes, $q<r$, $r \not\equiv 1$ (mod $q$), $qr<p$. Show $G$ has an unique Sylow $p$-subgroup.

I'm stuck on this exercise: Let $G$ be a group, with $|G|=pqr$, $p,q,r$ different primes, $q<r$, $r \not\equiv 1$ (mod $q$), $qr<p$. Show $G$ has an unique Sylow $p$-subgroup $P$. What ...
1
vote
2answers
81 views

Let $G$ be a finite group and $H < G$. Prove that $n_p(H) \le n_p(G)$.

Let $G$ be a finite group and $H < G$. Prove that $n_p(H) \le n_p(G)$. Ok, so now i know that $n_p(H) \le n_p(G)$ refers to the number of Sylow p-subgroups in H and G, respectively. From here, I ...
2
votes
1answer
39 views

Find the number of Sylow $p$-subgroups of $G$, if we know that $\lvert G\rvert=6$

Today I've been looking the third Sylow theorem. My professor did an example in class, so I tried to solve the example by myself and then compare what I did with the answer of my professor. The ...
0
votes
1answer
36 views

Why is the group abelian?

Lets say a group $G$ consists of 3 Sylow groups, $H_1,H_2,H_3$. Each of order $p_1,p_2,p_3$, that are prime numbers and different. Since we only have one of each Sylow group for each p, the second ...
1
vote
1answer
63 views

Let $P$ be a Sylow p-subgroup of a finite group $G$ and let $H$ be a normal subgroup of $G$. Show if $p$ does not divide $[G:H]$, then $P \subseteq H$

Let $P$ be a Sylow p-subgroup of a finite group $G$ and let $H$ be a normal subgroup of $G$. Show that if $p$ does not divide $[G:H]$, then $P \subseteq H$ I do not understand how to work this ...
2
votes
1answer
101 views

About first Sylow Theorem proof

I've been struggling with the first Sylow theorem proof we were given in class. This is how my professor introduced us the theorem: First Sylow Theorem: Let $G$ be a finite group. Let $\rvert G ...
1
vote
2answers
61 views

Let G be a group of order $p^2q^2$ where $p > q$ are primes. If $|G| \ne 36$, show that G has exactly one Sylow p-subgroup

Let G be a group of order $p^2q^2$ where $p > q$ are primes. If $|G| \ne 36$, show that G has exactly one Sylow p-subgroup. Ok, I'm not exatly sure where to begin with this one. I know that we are ...
2
votes
2answers
57 views

Let $S$ be a normal p-subgroup of a finite group $G$. Prove that $S \subseteq P$ for every Sylow p-subgroup $P$ of $G$

Let $S$ be a normal p-subgroup of a finite group $G$. Prove that $S \subseteq P$ for every Sylow p-subgroup $P$ of $G$. Now, I know that this involves the Sylow Theorems, of course. This is very new ...
3
votes
1answer
69 views

Exercise 257 from Rose“ A course on group theory”

I would please like some help on the exercise 257 of the Rose Group Theory. Let $G$ be a finite group and $H$ and $K$ be normal subgroups of $G$, and $P$ a Sylow p-subgroup of $G$. Then $(PH)\cap ...
6
votes
1answer
45 views

Let $G$ a finite group such that $\lvert G \rvert=pm$, with $p$ a prime and $\gcd(p,m)=1$. $G$ has an unique Sylow $p$-subgroup $P$. Prove $P\lhd G$.

I just made this exercise, left as homework, and I'm almost sure that I did something wrong, or at least that there's a better way to solve it. Here it goes: Let $G$ a finite group such that ...
5
votes
2answers
46 views

How to use Sylow theorems to prove that a subgroup of $SL_2(\mathbb{F}_3)$ is normal.

I have $G = SL_2(\mathbb{F}_3)$ and $H = \langle i, j \rangle$ where $$ i = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}, \quad j = \begin{pmatrix} -1 & 1 \\ 1 & 1\end{pmatrix} $$ (In ...
1
vote
3answers
45 views

Does $P \in Syl_p(N_G(P))$ implies $P \in Syl_p(G)$?

In a finite group $G$, let $P \in Syl_p(N_G(P))$, i.e. $P$ is a Sylow $p$-subgroup in its normaliser, does this imply $P \in Syl_p(G)$, i.e. it is a Sylow $p$-subgroup in its entire group?
2
votes
2answers
41 views

Sylow Subgroups of a Dihedral Group

Let $G = D_{2n}$ and $p$ an odd prime dividing $2n$, i.e. $2n = 2m\cdot p^{\alpha}$ with $p \nmid 2m$. I need to show that $P \in \operatorname{Syl}_p(G)$ is normal in $G$ and cyclic. As far as ...
1
vote
3answers
69 views

A group of order $1645$ is cyclic

This is what I have so far, but I get stuck at one point. $|G| = 1645 = 5\cdot 7\cdot 47$ I found with the Sylow theorem that there is a unique Sylow $5$-Subgroup, a unique Sylow $7$-Subgroup and ...
1
vote
1answer
53 views

Sylow theory question

I am asked: Does a group $G$ of order 14 have to be cyclic? Using Sylow's third theorem (work omitted), I conclude that $$|\textrm{Syl}_2(G)|=1 \textrm{ or }7$$ and $$|\textrm{Syl}_7(G)|=1$$ I ...
0
votes
1answer
69 views

A finite group of order $pq$ cannot be simple.

Let $p$ and $q$ be prime numbers. I wish to prove that a finite group $G$ of order $pq$ cannot be simple. Proof. Case 1: $p\not= q$. Case 2: $p=q$. Consider the first case where $p\not= q$. Without ...
0
votes
1answer
33 views

Groups and Characters exercise hint

I am trying to make my way through Grove's Groups and Characters but am having some trouble with the following seemingly benign exercise: If a group $G$ has a normal $p$-complement $K$ show that ...
0
votes
2answers
51 views

How do they know that the Sylow 3 subgroups are cyclic?

I have this exercise: Find all Sylow 3 subgroups of $S_4$ and demonstrate that they are all conjugate. This is the answer given in the manual: Now, Sylows first theorem says that every Sylow 3 ...
0
votes
0answers
51 views

exercise Sylow theorem

I have an exercise regarding Sylow theorems. I am supposed to add a number where it is blank. A Sylow 3-subgroup of a group of order 12 has order...?(the answer in the appendix is given to be 3) ...
3
votes
1answer
54 views

Sylow subgroup of a group

Let $G$ be a group such that $\vert G\vert=231$. I have to show that the unique Sylow 11-subgroup of $G$ is contained in the center of $G$ I proceed as follows: Since the number of Sylow 11-subgroup ...
1
vote
2answers
31 views

Why $n_i=j$ implies that there is a subgroup $H$ of $G$ of index j?

Why $n_i=j$ implies that there is a subgroup $H$ of $G$ of index j ? If I denote with $n_i$ the number of i-Sylow subgroups For example if $|G|=180=2^2\cdot3^2\cdot 5$ then if I assume that ...