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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24 views

Sylow theorems and normality

Let |G|=14924=2^2.7.13.41 We can see Sylow 41-subgroup is normal But when I try for Sylow 7 and 13, I face some problems. I use the argument following: Let P be a Sylow 41 and R be a Sylow 7 ...
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0answers
21 views

Let $G$ be a $p$-group, with $|G|= p^n$. Show that $G$ has a normal subgroup of order $p^m$ for each integer $0 < m < n$. [duplicate]

I think I have solved a problem using one of the sylow theorems. But, if this proof is correct, I think I've cheated a little. Since the chapter on Sylow theorems comes directly after the chapter on ...
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0answers
29 views

Why is a nontrivial finite group is nilpotent if every maximal subgroup is normal?

In the following proof I understand the above proof up to the part where it says "by Sylow theory $N(M)=M$", could someone explain to me why is this true. We have just started learning about group ...
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0answers
13 views

Injective map from the Cartesian product of two sylow-p-subgroups into the group.

Let $G$ be a group of order 148. Show that $G$ is not simple. The given solution goes as follows: $148 = 4 × 37$. By Sylow’s theorem, it has at least one subgroup $P$ of order 37. If $P'$ is ...
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1answer
15 views

The order of the normalizer of a $p$-subgroup of $S_{p}$ [closed]

I found it In Exercise in abstract algebra by Dummit and Foote. Let $P$ be a Sylow $p$-group of $S_p$. What is the order of $N_{S_p}(P)$?
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1answer
32 views

Groups of order 12 with a normal 3-subgroup contain an element of order 6

Let $G$ be a group of order $12$ with a normal $3$-subgroup (which is unique by Sylow's theorems). Does it contain an element of order $6$? I just need a hint to prove it without classifying all the ...
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2answers
31 views

Are the normalizers of Sylow p-subgroups isomorphic?

Let $G$ be a finite group and $A,B \in \text{Syl}_p G$, for some prime $p$. Is it always true that the normalizers $N_G(A)\cong N_G(B)$? I just need a hint to get started, because I don't know where ...
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1answer
86 views

Group of order $p^2+p $ is not simple [duplicate]

Can some one please give me a hint to prove that every group of order $p^2+p$ is not simple?
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1answer
31 views

Cardinality of a power set? Or is “all subsets of a set” $\neq$ power set?

Again as usual, group theory is muddling me up. A proof of the Sylow $I$ theorem starts as follows Let $X$ be the set of all subsets of $G$ with $|A|=p^m$. where, the setting I have for Sylow ...
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2answers
33 views

Non-abelian, simple subgroups of $S_n$

I am trying to prove, as part of a larger theorem, that if $G$ is a non-abelian finite, simple group of order $>2$ and $G$ is a subgroup of $S_n$, then $G$ must be a subgroup of $A_n$. Any ideas ...
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1answer
75 views

Classify groups of order 100 [closed]

So I am currently trying to Classify all groups of order 100 through an extensive proof; and this is as far as I have gotten so far, wondering how to go beyond the fact that both squares (Z4 & ...
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0answers
60 views

When can a dihedral group $D_{n}$ of order $2 n$ be a $p$-group?

A $p$-group is a group where the order of every group element is a power of the prime $p$. The presentation of a dihedral group $D_n$ of order $2 n$ is as follows. $$D_n = \langle x, y \mid x^n = y^2 ...
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1answer
51 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, ...
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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 ...
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0answers
55 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 ...
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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 $P = P_1 \times P_2$ ...
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1answer
33 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
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0answers
11 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 $X$ ...
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2answers
41 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 ...
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0answers
38 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 ...
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1answer
29 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 $G$?
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1answer
40 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 ...
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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 $P \in Syl_p(G)$, ...
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33 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 $H$?
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1answer
47 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 ...
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0answers
21 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?
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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 ...
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2answers
46 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 ...
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2answers
31 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 ...
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1answer
25 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 ...
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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 ...
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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 ...
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1answer
23 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 $|H|\geq 1+4\times 6$? ...
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25 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 ...
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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 ...
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1answer
123 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 ...
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2answers
46 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 $N\trianglelefteq S$, ...
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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 ...
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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 ...
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2answers
73 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 ...
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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 $S_2 \rtimes S(12)$. ...
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2answers
65 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, ...
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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 ...
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1answer
51 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
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1answer
29 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 ...
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0answers
49 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
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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 ...
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0answers
27 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 ...
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35 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 ...
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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 $a$. ...