For questions about Sylow theorems in the context of group theory.

learn more… | top users | synonyms

8
votes
2answers
92 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
48 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
39 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
41 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
0answers
42 views

For $G$ a group of order $1925$, find the number of Sylow $5$-subgroups

Let $G$ be a finite group of order $1925$. Find the number of Sylow $5$-subgroups in $G$. There must be $1$ or $11$ such subgroups. What is the actual number?
5
votes
2answers
103 views

Understanding what the Sylow theorems say about $p$-groups

I have a simple question. If we consider a group $G$ with order $p^k$ for a prime $p$. For example $125=5^3$. What we can obtain from sylows theorem? (I already understood it for the other cases, ...
1
vote
2answers
24 views

Group with order $|G|=p^2q^2$ and $p\not\mid q-1, p\not\mid q+1$

Group with order $|G|=p^2q^2$ and $p\not\mid q-1, p\not\mid q+1$. $q\not=p$ both prime. I want to show that there is only one $p$-Sylow subgroup. Let $S_p(G)$ the number of $p$-Sylow subgroups. I ...
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 ...
2
votes
1answer
40 views

Proof of Cauchy's Lemma in the case that G is abelian

I want to prove Cauchy's Lemma for abelian groups: If $G$ is abelian and there exists a prime such that $p$ divides the order of $G$, then there exists a $g \in G$ such that $p=\mathrm{ord}(g)$ I am ...
3
votes
1answer
60 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 ...
1
vote
1answer
96 views

Group of order 1183 is abelian if and only if contains an element of order 91

Let $G$ be a group such that $|G|=1183=7\cdot 13^2$. Show that $G$ is abelian if and only if $G$ has an element of order $91=7\cdot 13$. What i did: $7||G|\Rightarrow \exists x\in G : |x|=7$ and ...
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 $ ...
5
votes
2answers
93 views

Are all Sylow 2-subgroups in $S_4$ isomorphic to $D_4$?

I was assigned to show that every Sylow 2-subgroups in $S_4$ is isomorphic to $D_4$. So I figured, since $|S_4|=24=2^3\cdot 3$, every Sylow 2-subgroup has either the form: $\langle ...
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 ...
1
vote
1answer
55 views

Conjugacy Classes of p-Sylow Subgroups

I've run into a brick wall with a problem in Pinter's Book of Abstract Algebra where it attempts to guide the reader through a proof of the first Sylow theorem; if I take the result as given, I can ...
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 ...
1
vote
1answer
34 views

Let $F$ be a free centralizer in $G$. Then $F=P\times A$, where $P$ is a Sylow $p$-subgroup of $F$ and $A$ is abelian.

This question is from the Proposition A.23.2 in page 520 of the book ``Berkovich, Yakov, and Zvonimir Janko. Berkovich, Yakov; Janko, Zvonimir: Groups of Prime Power Order. Vol. 2. Walter de Gruyter, ...
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 $Q=\phi(P)$.
1
vote
1answer
45 views

trouble applying Sylow's theorems

Let G be a simple group and let n_p be the number of Sylow p-subgroups, p prime. Show that |G| divides (n_p)! (factorial). If i start off by assuming G is abelian then G is isomorphic to Z/pZ. So |G| ...
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
98 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 ...
0
votes
1answer
36 views

Proving group is $p$-group by contradiction

http://www.proofwiki.org/wiki/Group_is_P-Group_iff_All_Elements_have_Order_Power_of_P Is $k$ a prime or a prime power? Sorry for this stupid question but I can't tell what $k$ is in this context ...
3
votes
1answer
53 views

Does Sylow's theorem assert the existence of subgroups of order $p^j$ for all $j=1,\dots,k$ ?

Sylow's theorem says that there exists a subgroup of order $p^k$, where $p^k$ is the highest power of $p$ dividing the order of the group. But for example if the group order is $24$, then we can ...
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 ...
2
votes
3answers
43 views

if $Q$ and $P$ are distinct $p$-Sylow subgroups then $Q\not\subseteq N_G(P)$.

I have been told to use the following to prove another claim, but I would like to prove this anyway for myself. However I can't tell why it's true. I think it's true, but can't see why! Here it is: ...
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 ...
2
votes
1answer
35 views

question about $p$-Sylow subgroups

I was wondering whether it is necessarily true that if $P_1$ and $P_2$ are Sylow $p$-subgroups of $G$ then $P_2\subseteq N_G(P_1)$. I don't think it is because since they're both Sylow $p$-groups, ...
1
vote
1answer
49 views

showing that a group of order 45 is abelian

I'm trying to understand the following proof from Dummit & Foote (pg. 137) which shows why a group of order 45 is abelian. I understand everything but the last two sentences. Why is it that ...
4
votes
2answers
88 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 ...
0
votes
0answers
12 views

group of order $p^n$contained in a $p$-Sylow subgroup?

Let $G$ be a finite group and $p$ a prime number. Let $\Sigma$ be the set of Sylow p-subgroups. Let $H$ be some subgroup of $G$ containing $p^n$ elements for some number $n$. The group $H$ acts on ...
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 ...