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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5
votes
1answer
38 views

Group with exactly six Sylow 5-subgroups?

Give an example of a group with exactly six Sylow 5-subgroups. I think $A_5$ works because it has 6 subgroups of order 5: $\langle(12345)\rangle,\langle(12354)\rangle, \langle(12435)\rangle, ...
2
votes
1answer
46 views

Square of order of a Sylow p-subgroup in the nonabelian simple groups

Is it true that for all Sylow subgroups $P$ of a nonabelian simple group $G$ that $|P|^2 < |G|$? If $P$ is abelian, this is an easy consequence of Brodkey's theorem (Suppose that a Sylow ...
0
votes
0answers
53 views

Number of $2$-Sylow subgroups of $S_5$

Find the number of $2$-Sylow subgroups of $S_5$ and represent one of them. Would someone please give a hint for how to start?! I only can say that it should be an odd number dividing $5!$ (these can ...
0
votes
0answers
44 views

I need help proving that a certain subgroup is a direct product of specific subgroups

Let G be a group, P be an abelian Sylow-p subgroup of G. Let $N=N_G(P)$ and assume that H is a complement of P in N which I believe means that $HP=N$ and $H\cap{P}=1$ Prove that $P = P_1 \times P_2$ ...
0
votes
0answers
23 views

If The order of $G$ is $pm$ and $p$ is prime such that If $H$ is normal to $G$ with order $p$ show that H is characteristic

I will be grateful for your help If The order of $G$ is $pm$ and $p$ is prime such that If $H$ is normal to $G$ with order $p$ show that H is characteristic
0
votes
0answers
10 views

A $p$-subgroup of a finite group is either a Sylow $p$-subgroup or properly containted in a Sylow $p$-subgroup of its normalizer

This exercice is from Aschbacher's book "Finite group theory". I am not asking for a complete solution, just for a hint. Here is a partial solution, when the ambient group $G$ is a $p$-group. If $X$ ...
0
votes
2answers
39 views

Does $S_6$ have an abelian sylow $2$ subgroup.

How do I check if $S_6$ has an abelian sylow 2 subgroup. Order of any sylow 2 subgroup is $16$ and by sylows theorem it has $45$ sylow 2-subgroups, but how to check whether it has any abelian sylow ...
1
vote
0answers
36 views

Show that the number of conjugates of a p-Sylow subgroup is not divisible by p

I'm trying to solve the following problem: "Let $P$ be a $p$–Sylow subgroup of a finite group $G$. Consider the set of conjugate subgroups $gPg^{-1}$ with $g \in G$. Show that the number of ...
1
vote
1answer
27 views

If $S$ is a sylow $p$-subgroup of $G$ and $H \lhd G$ and $S\subset H$, how to show that if $S$ is normal in $H$ then its normal in $G$?

Let $G$ be some finite group. If $S$ is a sylow $p$-subgroup of $G$ and $H$ normal subgroup of $G$ and $S\subset H$, how to show that if $S$ is normal in $H$ then its normal in $G$?
2
votes
1answer
39 views

What is the difference between a $p$-group and a Sylow group?

We have just started a course about Group theory. I am confused about the difference between a $p$-group and a Sylow group. As I understand it, a group $G$ is called $p$-group if $|G|=p^{m}$, where ...
0
votes
0answers
25 views

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 $P \in Syl_p(G)$, ...
0
votes
0answers
31 views

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 $H$?
1
vote
1answer
38 views

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 ...
0
votes
0answers
20 views

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?
0
votes
0answers
32 views

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 ...
3
votes
2answers
45 views

$G$ 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 ...
0
votes
2answers
29 views

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 ...
1
vote
1answer
23 views

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 ...
2
votes
0answers
20 views

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 ...
1
vote
1answer
55 views

Sylow subgroup of some factor group.

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 ...
1
vote
1answer
22 views

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 $|H|\geq 1+4\times 6$? ...
1
vote
0answers
24 views

Calculating amount of $2$-sylow subgroups of $S_{2^n}$.

Main question: How do I calculate the number of $2$-sylow subgroups of $S_{2^n}$? Let $n \in \mathbb{Z}_{\geq 2}$. I have a $2$-sylow subgroup $H \subset S_{2^n}$ (too long to spell out all the ...
0
votes
1answer
31 views

Let $G$ a finite group and $p$ a prime. If $P$ is the unique p-Sylow of $G$ and $f: G \to G$ is an homomorphism, then $f(P) < P$

Let $G$ a finite group and $p$ a prime. If $P$ is the unique p-Sylow of $G$ and $f: G \to G$ is an homomorphism, then $f(P) < P$. Well, as $P$ is the unique p-Sylow of $G$, $P$ is a normal ...
2
votes
1answer
116 views

Does $G$ contain a normal subgroup that is either a 2-group, 3-group, 5-group or of order a multiple of 15?

Suppose that a group $G$ of order 450 has exactly two subgroups, $H$ and $K$, of order 225. Show that $G$ contains a non-trivial normal subgroup that is either a 2-group, a 3-group, a 5-group ...
2
votes
2answers
45 views

A normal $p$-group of $G$ is contained in each Sylow subgroup

I have shown that if $S\in \text{Syl}_p(G)$ and $N\trianglelefteq G$, then $N\cap S\in \text{Syl}_p(N)$. After that I am asked to show that if $N$ is also a $p$-group then $N\trianglelefteq S$, ...
3
votes
1answer
59 views

Do we conclude in that way that it is a $p$-Sylow subgroup?

I am looking at the following exercise: If $G$ is finite and $f:G\rightarrow H$ is a group epimorphism, show that if $P\in \text{Syl}_p(G)$ then $f(P)\in \text{Syl}_p(H)$. $$$$ I have done the ...
2
votes
1answer
68 views

Show that there is such a Sylow subgroup

I want to show that if $G$ is finite and $f:G\rightarrow H$ is a group epimorphism and if $Q\in \text{Syl}_p(H)$ then there is a $P\in \text{Syl}_p(G)$ with $Q=f(P)$. $$$$ I have done the ...
2
votes
2answers
72 views

Show that if $G$ is finite and $f:G\rightarrow H$ is a group epimorphism, $|\text{Syl}_p(G)|\geq |\text{Syl}_p(H)|$.

I want to show that if $G$ is finite and $f:G\rightarrow H$ is a group epimorphism, $|\text{Syl}_p(G)|\geq |\text{Syl}_p(H)|$. $$$$ I have done the following: Since $f$ is a group epimorphism we ...
0
votes
0answers
17 views

Sylow $2$-subgroup of $A^m \rtimes S_m$

Let $G=A^m \rtimes S_m$ where $A$ is some abelian group. Now what can I say about sylow $2$-subgroup of $G$. The text I am reading says let $S$ be the fixed sylow $2$-subgroup $S_2 \rtimes S(12)$. ...
2
votes
2answers
61 views

Show $N_G(N_G(P))=N_G(P)$ when $P$ is a Sylow $p$-group.

I am looking at the following exercise: Let $P$ be a $p$-Sylow subgroup of $G$ then $N_G(N_G(P))=N_G(P)$. When $P\in \text{Syl}_p(G)$ it holds that $P\leq N_G(P)$, or not? If this is true, ...
0
votes
1answer
22 views

Let A be a finite group and P be a normal p sylow subgroup. What is the connection between P and $Tor_p(A)$

Let A be a finite group and P be a normal p sylow subgroup. can there be an element $g \in A$ where $order(g) = p^x$ where x>0 and $g \notin P$ ? what I really try to understand is the connection ...
0
votes
1answer
50 views

The group is not simple

I want to show that if $|G|=pqr$ where $p,g,r$ are primes, then $G$ is not simple. We have that a group is simple if it doesn't have any non-trivial normal subgroups, right? $$$$ I have done the ...
2
votes
1answer
28 views

Some properties of a finite group with all Sylow subgroups that are cyclic

I consider a finite group $G$ such that all its Sylow's subgroups are cyclic. I suppose that $|G|=p_1^{k_1}...p_n^{k_n}$ with $p_1<...<p_n$ distinct primes. Can I say something about the ...
0
votes
0answers
48 views

Use intersection of Sylow subgroup to prove no simple group of order 525

Assume there exists a simple group with order $525$ $525 = 3*5*5*7$ we get $n_3 \in \{1, 5, 7, 25, 35, 150\}, n_5 \in \{1, 21\}, n_7 \in \{1, 3, 5, 15\}$ Assuming the group is simple $n_5 = 21, n_7 ...
4
votes
1answer
62 views

Show that $G$ is abelian.

Let $G$ be a finite group such that every Sylow subgroup of $G$ is normal and abelian.Show that $G$ is abelian. Let $x,y\in G$ . Case 1:If $x,y$ are in the same Sylow subgroup and as it is given to ...
0
votes
0answers
26 views

Clarification : No simple group for order $5103 = 3^6 \cdot 7$

No simple group for order $5103 = 3^6 \cdot 7$ using the small index argument and assuming the group is simple: consider $n_{3}$ this must divide the rest of the order of the group and be congruent ...
0
votes
0answers
34 views

*ADVICE* Intuition about techniques for disproving the existence of simple groups of given order

Looking for some feedback on how to know when to use the different techniques for disproving there are no simple groups of a given order. the techniques being: 1) counting elements 2) using ...
1
vote
3answers
53 views

Group with at least 2 subgroups of order $p$ has at least $p^2-1$ elements of order $p$.

Let $G$ be a finite group and $p$ be a prime number. Let $a,b$ be two elements of order $p$ such that $b\notin \langle a\rangle $ where $\langle a\rangle $ denotes the subgroup generated by $a$. ...
0
votes
1answer
55 views

Two conjugated groups of an article.

Working in a proof given by Broto, given a finite group $G$, a Sylow $p$-subgroup $S$, a $p$-group $P\leq S$ a group $gPg^{-1}\leq S$ with $g\in G$ such that $C_S(gPg^{-1})$ is a Sylow $p$-subgroup of ...
0
votes
1answer
30 views

Let $G$ be a group of order $p^2q^2$ , $p$ does not divide $q^2-1$ , $q$ does not divide $p^2-1$ , then is $G$ abelian? [closed]

Let $G$ be a group of order $p^2q^2$ ,where $p,q$ are primes , $p$ does not divide $q^2-1$ ,and $q$ does not divide $p^2-1$ , then is $G$ abelian ?
1
vote
1answer
35 views

Let a finite group $G$ have $n(>0)$ elements of order $p$(a prime) . If the Sylow p-subgroup of $G$ is normal, then does $p$ divide $n+1$?

Suppose $G$ is a finite group and $p$ is a prime that divides $|G|$. Let $n$ denote the number of elements of $G$ that have order $p$ . If the Sylow p-subgroup of $G$ is normal, then is it true that ...
0
votes
3answers
54 views

Special linear group of order 2 over field of order 3

Let G = SL(2, F$_3$) (group of matrices of determinant 1 over the field of order 3). Find |G|. Show that Z(G) is not {1$_G$}. Determine the number of Sylow 3-subgroups of G. What is the isomorphism ...
0
votes
2answers
30 views

When $|G|=105$ and has a normal Sylow $3-$subgroup, then $G$ is abelian.

$|G|=105=3\cdot 5\cdot 7$. We know that the Sylow $5-$subgroup $P_5$ must be normal and the Sylow $7-$subgroup $P_7$ must also be normal by simply counting elements. With the additional assumption of ...
1
vote
3answers
77 views

Elements of order 2 in the special linear group

I am trying to show that the unique Sylow $2$-subgroup of the special linear group $SL(2,\mathbb{F_{3}})$ is isomorphic to the quaternion group $Q_{8}$. Call the unique Sylow $2$-subgroup $P$, and ...
1
vote
1answer
80 views

Number of Sylow 3-subsgroups of special linear group

I am aware that questions on this topic are around on this site, but they all seem to require information about the group that is not available to me in this problem. Consider the special linear ...
1
vote
2answers
60 views

Prove that no group of order $p^2q$ is simple where $p$ and $q$ are prime

Can we argue: There is a Sylow $p$-subgroup $K$ of order $p^2$ which should be abelian and thus normal and so $K$ is a non trivial normal subgroup
5
votes
0answers
43 views

A basic query on sylow subgroup

Let $G$ be a group of odd order then $2 \not\mid\ \lvert G \rvert$, so can we say that $G$ has a sylow $2$- subgroup which is $\{e\}$ or a sylow $p$-subgroup of $G$ is only defined if $p \mid \lvert ...
0
votes
0answers
39 views

Sylow counting to show group is isomorphic to semidirect product

Let G be a group of order $|G|=pq^m$ where $p, q$ are primes with $q^m<p$. Use a Sylow counting argument to show that $G\cong C_p \rtimes_h Q$ where $Q$ is a group with $|Q|=q^m$ and ...
2
votes
0answers
67 views

$G$ be a finite group and $f \in Aut (G)$ such that $f^3$ is identity and $f$ has unique fixed point , then any $p$-Sylow subgroup is normal?

Let $G$ be a finite group and $f \in Aut (G)$ such that $f^3$ is identity and $f(x)=x \implies x=e$ ; then is it true that for every prime $p$ dividing $|G|$ , there is exactly one $p$-Sylow subgroup ...
1
vote
1answer
65 views

Sylow counting argument; prove G isomorphic to the direct product.

Let G be a group of order $|G|=pq^m$, where $p$ and $q$ are primes with $q^m<p$. i) Use a Sylow counting argument to show that $G\cong C_p\rtimes_hQ$ where Q is a group with $|Q|=q^m$ and ...