8
votes
2answers
85 views

A group of order $120$ has a subgroup of index $3$ or $5$ (or both)

What I have tried that number of $2$-sylow subgroup can be $1,3,5$ or $15$.I have solved the problem when the number of $2$-sylow subgroup is $1,3,5$. But I am not able to solve it for $15$. Any help ...
0
votes
2answers
47 views

Number of non isomorphic groups of order $122$, My attempt through Sylow theory.

$|G|=122 = 2 . 61$ No. of sylow $2$ subgroups $= 1$ or $61 = n_2$ No. of sylow $61$ subgroups $= 1 = n_{61}$ Let the group of order $61$ be $H_{61}$ and the group of order $2$ be $H_2$ Then : ...
2
votes
2answers
28 views

If $H$ is a normal subgroup of a finite group $G$ and $|H|=p^k$ for some prime $p$. show that $H$ is contained in every sylow $p$ subgroup of $G$

If $H$ is a normal subgroup of a finite group $G$ and $|H|=p^k$ for some prime $p$. show that $H$ is contained in every sylow $p$ subgroup of $G$ Attempt: $|H|=p^k \implies |G|=p^{n_1} q^{n_2} ...
1
vote
2answers
38 views

Show that if $G$ is a group of order $168$ that has a normal subgroup of order $4$ , then $G$ has a normal subgroup of order $28$

Show that if $G$ is a group of order $168$ that has a normal subgroup of order $4$ , then $G$ has a normal subgroup of order $28$. Attempt: $|G|=168=2^3.3.7$ Then number of sylow $7$ subgroups in $G ...
2
votes
1answer
40 views

Smallest possible odd integer that can be the order of a non-Abelian group.

Smallest possible odd integer that can be the order of a non-Abelian group. Attempt: A non abelian group means $Z(G) \subset G$ . Hence, it suffices to find the smallest odd integer $n$ such that ...
2
votes
1answer
49 views

The smallest composite integer $n$ such that there is a unique group of order $n$?

We need to find the smallest composite integer $n$ such that there is a unique group of order $n$. Attempt: Let us suppose that $n= ab$ is a composite integer where $a,b$ are integers such that ...
2
votes
1answer
43 views

The only group of order $255$ is $\mathbb Z_{255}$ ( Using Sylow and the $N/C$ Theorem)

Let $G$ be a group such that $G =255 = 3.5.17$. Let $H$ be a sylow $17$ sub group of $G$. Then, by Sylow's theorem : the number of sylow $17$ sub groups can be $n=1,18,35,...$ . Since, $n$ should ...
1
vote
1answer
30 views

Let $G$ be a finite group and let $a$ be an element of $G$. Then, $|cl ~(a)| = |G:C(a)|$

Let $G$ be a finite group and let $a$ be an element of $G$. Then, $|cl ~(a)| = |G:C(a)|$ where $cl ~(a)$ refers to the conjugacy class of $a$. The proof in the book which I am reading asks to prove ...
0
votes
2answers
64 views

Use Sylow's theorem to show that $G = HN_G(P)$

I am stumped on this question. Does anyone have some helpful hints or a solution to this question? Thanks! Let $G$ be a group of finite order. Let $H$ be a normal subgroup of $G.$ Let $P$ be a ...
3
votes
1answer
59 views

Proving that the intersection of a Sylow p-group with a normal subgroup is also a Sylow p-group

An exercise from Dummit and Foote pg. $101$ ex. $9$ asks to show the following: Let $G$ be a group of order $p^{a}n$ where $p$ does not divide $n$ and let $N\unlhd G$ so that $|N|=p^{b}m$ where ...
0
votes
1answer
49 views

Inconsistent definition of Sylow p-subgroup

Here is the definition of a Sylow $p$-subgroup from Wikipedia: For a prime number $p$, a Sylow $p$-subgroup (sometimes $p$-Sylow subgroup) of a group $G$ is a maximal $p$-subgroup of $G$, i.e., a ...
-1
votes
1answer
86 views

Elements whose orders are multiple of $p$ [closed]

Let $G$ be a non-solvable group, $N$ an abelian minimal normal $p$-subgroup of order $p^r$ with $p\notin \pi(G/N)$, $N=C_G(N)$ and $K=G/N\cong A_5$. By these assumption we can conclude that $G$ has ...
3
votes
1answer
102 views

Computing values of centralizers in a non-solvable group with a given property

A finite group G satisfies property $P_n$ if for every prime integer $p$, $G$ has at most $(n−1)$ non-central conjugacy classes the order of the representative element of which is a multiple of $p$. ...
7
votes
1answer
138 views

A question about Sylow subgroups

Let $G$ be a finite group and $P\neq\{e\}$ be a Sylow $p$-subgroup of $G$ and $P^g\neq P$ be its conjugate in $G$. If we know that $P\cap P^g\neq \{e\}$, can we conclude that $Z(P)\cap Z(P^g)\neq ...
4
votes
2answers
142 views

Is a finite group always a element-wise product of Sylow subgroups?

Let $G $ be a finite group, and let $ p_1, \ldots, p_n $ be the distinct primes dividing $|G|$. For each $i $, let $ P_i $ be a Sylow $ p_i $-subgroup of $ G $. I seem to recall a theorem saying $ ...
1
vote
3answers
39 views

How many Sylow-$ 3$ subgroup does $G$ have?

Let $G$ be a noncyclic group of order $21$. How many sylow-$3$ subgroup does G have? The possible orders of Sylow $3$ subgroups is $1, 7$. But how to check the exact number?
2
votes
0answers
37 views

Neccessary Condition involving Sylow-Subgroups for $p$-Solvability

Let $G = AB$ and let $P \unlhd A$, where $P \in Syl_p(G)$. If $P$ permutes with all Sylow $q$-subgroups of $B$ with $q \ne p$, then $G$ is $p$-solvable. Any suggestions on how to proof?
0
votes
1answer
25 views

Sylow-Subgroups and arbitrary groups where their order contains the same prime-power.

Let $|G| = p^k m$ with $p$ and $m$ being coprime. Then it is well known that there exists a subgroup $S$ of $G$ with $|S| = p^k$, the so called Sylow-$p$-subgroups. Now let $U \le G$ be some subgroup ...
3
votes
0answers
32 views

Simple question on Sylow subgroups.

Given a finite group $G$, $|G|=p^am$, $P,Q\in\operatorname{Syl}_p(G)$, then we know that $Q=P^x$ for some $x\in G$. Thus every element of $Q$ is the image of an element of $P$ under the isomorphism ...
0
votes
0answers
32 views

On the possible subgroups of a Sylow subgroup

Let $G$ be a finite group, say $|G|=p^am$, with $(p,m)=1$, Let $n$ be the number of $p$-Sylow subgroups of $G$. Call them $P_1,\dots,P_n$. Is true that every subgroup of $G$ of order $p^b$ with $b\le ...
2
votes
1answer
46 views

Question about Sylow Theorem and normalizer

I'm dealing with the following problem. Let $G$ be a finite group, $H$ and $K$ Sylow 3- 5- subgroups respectively of $G$. Suppose that 3 divides $|N(K)|$, show that 5 divides $|N(H)|$. I've ...
0
votes
1answer
47 views

Subgroups of a group of order 60 with a normal subgroup of order 2 (Sylow)

This is the problem 38 of the chapter 24 in the Gallian's Algebra. Suppose that $G$ is a group of order $60$ and $G$ has a normal subgroup $N$ of order $2$. Show that: $G$ has normal ...
1
vote
3answers
27 views

$P\in Syl_p(G)\Rightarrow|G:N_G(P)|=|Syl_p(G)|$

Let $G$ be a finite group s.t. $|G|=p^rm$, where $(p,m)=1$. Let then $P$ be a $p$-Sylow subgroup of $G$, i.e. $P\le G$ with $|P|=p^r$. We want to show that $|G:N_G(P)|$ is the number of $p$-Sylow ...
2
votes
1answer
74 views

Question about $p$-Sylow subgroups of the quotient group

I have been working on the following problem. Let $G$ be a finite group, $N\trianglelefteq G$ and $p$ a prime, then $n_{p}(G/N)\leq n_{p}(G)$. I have beeen trying to solve it, but it seems I ...
0
votes
1answer
46 views

Sylow questions on $GL_2(\mathbb F_3)$.

Consider $G:=GL_2(\mathbb F_3)$. I have to extrapolate as much information about it as I can. Without computations. First of all: I think someone else has already done this before me, hence if you ...
0
votes
3answers
68 views

If Q is a p-Sylow-Group of H there is a p-Sylow-Group P of G with $\phi(P)=Q$ while $\phi:G\rightarrow H$ epimorphism

Let G be a finite group and $\phi: G \rightarrow H$ a group-epimorphism. Proof: If $Q\in Syl_p(H)$ there is a $P\in Syl_p(G)$ with $Q=\phi(P)$.
4
votes
1answer
54 views

Question about Sylow $p$-subgroups

If a group $H$ has order $255$ then the Sylow theorems tell us that it must have a Sylow $p$-subgroup of order $5$ and there are either $1$ or $51$ of them, also there is either $1$ Sylow $p$-subgroup ...
5
votes
2answers
87 views

How many nonabelian groups of order 2009? (Check work)

I just need someone to check this argument. Let $G$ be a nonabelian group of order $2009$. The prime factorization of $2009$ is $7^2 \cdot 41$. Let $n$ be the number of Sylow 7-subgroups. Then $n ...
3
votes
1answer
97 views

A Group of Order $540$ is not simple

Why is a group of order $540$ not simple? The hints I have been given are not helpful. Here's what I have been told. Let $G$ be such a group. Then there are $36$ Sylow $5$-subgroups; let $H$ be ...
2
votes
2answers
98 views

This is about Sylow subgroups of Alternating group $A_n$ (Multiple choice)

This is a question from a competitive exam. For a positive integer $n\ge 4$ and a prime number $p\le n$ denote $U_{p,n}$ to be the union of all $p$-sylow subgroups of alterbating group $A_n$. Also ...
1
vote
0answers
37 views

Fixed Points and Sylow-Subgroups of Subgroups who are also Sylow-Subgroups of whole Group

Let a finite group $G$ act on a set $\Omega$. For some $\alpha \in \Omega$, denote by $G_{\alpha} = \{ g \in G : \alpha^g = \alpha \}$ the stabiliser of $\alpha$ in $G$. I want to show that: (i) if ...
4
votes
2answers
87 views

What do Sylow 2-subgroups of finite simple groups look like?

What do Sylow 2-subgroups of finite simple groups look like? It'd be nice to have explanations of the Sylow 2-subgroups of finite simple groups. There are many aspects to the question, so I envision ...
2
votes
1answer
23 views

Question about a non-abelian group of order $p^2q$

Suppose $p<q$, where $p,q$ are primes and we have a non-abelian group $G$ of order $p^2q$. Is it true that it has a subgroup which is not normal? I try to use Sylow's theorems. We take Sylow ...
2
votes
1answer
55 views

A question about the involution in simple groups.

Let $G$ be a finite simple group of Lie type over a finite field ($F_q$) of order $q$ with characteristic $p\neq 2$. Suppose $S$ is $2$-sylow subgroup of $G$ and is not abelian. I have two question ...
1
vote
2answers
117 views

Sylow $7$-subgroup of a group of order $4\cdot3\cdot5\cdot7$ is normal

How to show that the sylow $7$-subgroup of a group of order $420$ is normal. I Know that it is true using GAP. But how to show it without using GAP. I don't know how to start this. Thanks for any ...
0
votes
1answer
33 views

Ruling out orders when applying Sylow's theorems

Going through examples of applications of the Sylow theorems in Fraleigh's book, when proving that no group of order 36 is simple, after concluding that $| H \cap K | = 3$ for two $3$-Sylows $H$,$K$, ...
7
votes
1answer
103 views

Show that if the prime $p$ divides $|G|$, then $|X|$ is divisible by $p$.

Question : Let $p$ be a prime number that divides the order of the finite group $G$. Let $X$ = $\bigcup_{P \in Syl_p(G)}P$. Show that $|X|$ is divisible by $p$.
1
vote
2answers
35 views

A Criterion for being Sylow p-group

Show that if $H$ is a $p$-group of finite group $G$ and $N_G(H)=H$ then $H$ is a Sylow $p$-group of $G$? Or prove the following more general property,$$[G:H]\equiv1\ (\mod\ p)$$
2
votes
2answers
94 views

An application of Sylow theorems in p-groups!

If $G$ is a finite group of order $p^{n}$ (which $p$ is a prime number) and have only one subgroup of order $p^{n-1}$ ,namely $H$ ,then $G$ is cyclic ! My "proof" is as follows: suppose $$x\in G-H$$ ...
0
votes
1answer
36 views

Normal Sylow $p$-subgroup of a normal subgroup

Any hints for the following question - I am sure that I am missing something very simple here. $K$ is a normal subgroup of $G$ and $P$ is a Sylow $p$-subgroup of $K$. If $P$ is a normal subgroup of ...
0
votes
0answers
23 views

Sylow subgroups and normalizers [duplicate]

Let $H$ be a Sylow 3-subgroup and $K$ a Sylow 5-subgroup of a finite group $G$. Suppose $|H|=3$ and $|K|=5$ and $N_G(K)$, has an element of order 3. Show that $N_G(H)$ has an element of order 5.
1
vote
1answer
32 views

$2$-Sylow subgroups of $S_{14}$

I need to describe what are the $2$-Sylow subgroups of $S_{14}$. I don't know how to start thinking about this since is it too large. I know that all that I must do is to find one of them since all ...
3
votes
3answers
132 views

Question about soluble and cyclic groups of order pq

I'm trying to solve the following problem: If $p>q$ are prime, show that a group $g$ of order $pq$ is soluble. If $q$ doesn't divide $p-1$, show that $pq$ is cyclic. Show that two non abelian ...
0
votes
1answer
97 views

Show that the p-Sylow subgroup is normal in $G$

Let $G$ be a finite group and suppose that $\phi$ is an automorphism of $G$ such that $\phi^3$ is the identity automorphism. Suppose further that $\phi(x) = x$ implies that $x = e$. Prove that for ...
1
vote
1answer
47 views

Prove $G$ has a normal Sylow subgroup

Let $|G|=pqr$ where $p, q$ and $r$ are prime and $p < q < r$. Prove that $G$ has a normal Sylow subgroup for either $p, q$ or $r$. Let $n_p, n_q, n_r$ denote the number of Sylow subgroups for ...
1
vote
1answer
44 views

Finite group $G$ is product of a subgroup $H$ and normalizer of a Sylow $p$-subgroup of $H$

Let $G$ be a finite group, $H$ a normal subgroup of $G$ and $P$ a Sylow $p$-subgroup of $H$. Let $N_G(P)$ be the normalizer of $P$ in $G$. Show that $G=N_G(P)H$.
0
votes
1answer
60 views

Sylow subgroups of soluble groups

Suppose $G \leqslant S_p$ acts transitively on $\{1,...,p\}$ for prime $p$. Let $P \leqslant G$ be a Sylow p-subgroup. Is it true that $G$ is soluble <=> $P \triangleleft G$?
0
votes
1answer
77 views

Normalizer of Sylow p subgroup

I am trying to prove a simple group of order 168 has no subgroup of order 14, the hint ask me show how many Sylow 7 subgroup it has and what is the order of the normalizer of such Sylow 7 subgroup. I ...
0
votes
1answer
52 views

Order of Sylow $p$-subgroups

My class is studying on Sylow $p$-subgroups, and I had been stuck for several hours on determining the order of a Sylow $p$-subgroup of a group $G$ of finite order. I asked a previous question like ...
8
votes
3answers
366 views

If H is a subgroup of G, then H has no more Sylow subgroups than G

If $H$ is a subgroup of the finite group $G$, then how do I show that $n_p(H) \leq n_p(G)$? Here $n_p(X)$ is the number of Sylow $p$-subgroups in the finite group $X$. Here is my attempt: Suppose ...