For questions about Sylow theorems in the context of group theory. Not for use with questions regarding Sylow systems, which belong in solvable-groups.

learn more… | top users | synonyms

1
vote
2answers
25 views

Sylow p-subgroup of order p does not normalize any other Sylow p-subgroup

Let $P_1,P_2$ be distinct Sylow p-subgroups of $G$ with order $p$. Is it generally true that $P_1$ cannot normalize $P_2$? I've seen algebra textbooks use this fact for $p=3,11$ and they quote 'a ...
0
votes
2answers
47 views

Group of an order 385

Let $G$ be a group of order $385$, proof that $Z(G)~~ (cent(g))$ contains object of order $7$. I used sylow theorm and realized that there are $1$ sylow-$11$ sub-group which is normal in $G$ and also ...
3
votes
1answer
37 views

$ S_{4} $ has a sylow tower?

A group having a Sylow tower is a finite group that possesses a Sylow tower: a normal series such that the successive quotient groups of the normal series all have orders that are powers of primes, ...
2
votes
0answers
41 views

$ p $ the largest prime factor of $ \vert G \vert $. I show if $ p > 3 $ and $ P \in Syl_{p}(G) $ then $ P \unlhd G $.

Let $ G $ be a soluble group, $ p $ the largest prime factor of $ \vert G \vert $. I show if $ p > 3 $ for prime $ p $ and $ P \in Syl_{p}(G) $ then $ P \unlhd G $. For proof i employ the ...
0
votes
1answer
15 views

$ G $ is $ p $-supersolvable group . $ Q \in \operatorname{Syl}_{2}(G^{\prime}) $. Show $ Q \unlhd G $.

Let $ G $ is a finite $ p $-supersolvable group for odd prime numbers . Suppose $ Q \in \operatorname{Syl}_{2}(G^{\prime}) $. Now i'll show $ Q \unlhd G $. Since $ G $ is $ p $-supersolvable, then $ ...
2
votes
2answers
54 views

Normalizer of a Sylow 2-subgroups of dihedral groups

I can't solve the following exercise which is the last exercise in page 146 of Dummi & Foote's Abstract Algebra: Let $2n=2^ak$ where $k$ is odd. Prove that the number of Sylow 2-subgroups of ...
1
vote
1answer
41 views

p-element centralizing a Sylow p-subgroup

Let $G$ be a finite group, $P$ a Sylow $p$-subgroup for a prime $p$ and $g$ a $p$-element with $gxg^{-1} = x$ for all $x \in P$. Then $g \in Z(P)$. Is this true? How can i prove that $g \in P$ ...
-2
votes
1answer
28 views

how many elements are there which has order 3?

if the number of sylow-3 subgroups of a group of order 96 is 4, how many element are there which has order 3? I dont know how to start. why isnt the answer 4?
5
votes
2answers
56 views

Show that if $|G| = 30$, then $G$ has normal 3-Sylow and 5-Sylow subgroups.

Show that if $|G| = 30$, then $G$ has normal $3$-Sylow and $5$-Sylow subgroups. Let $n_3$ denote the number of 3-Sylow subgroups and $n_5$ the number of $5$-Sylow subgroups. Then, by the third ...
0
votes
1answer
40 views

Find Sylow $2$-subgroups of polyhedron

The question is to find Sylow $2$-subgroup of 1) T(Tetrahedron) 2) O(Octahedron) 3) I(Icosahedron) From the Sylow theorem, I get the order of Sylow 2-subgroup in T($2^2\times3$), ...
3
votes
2answers
54 views

for prime $q\ge p$, integer $m\ge 0$, any group with order $p^2 q^m$ is not simple

I tried the two ways, but both are all failed. First, counting the order of union of all Sylow $p$, $q$-subgroup. Second, group action from orginal group to set of cosets by Sylow subgroup. Is there ...
2
votes
1answer
45 views

Problem involving Sylow Theorem from Michael Artin's book.

Let $n = p^em$, and let $N$ be the number of subsets of order $p^e$ in a set of order $n$. Determine the congruence class of $N$ modulo $p$. I think the answer is m and I feel it is a little similar ...
2
votes
0answers
43 views

Why sylow $ p $-subgroups of $ H $ are Sylow $ p $-subgroups of $ G $?

Let $ H $ is a subgroup of $ G $ that $ \vert G : H \vert $ is a $ \pi $-number and there exist a nilpotent subgroup $ K $ of $ G $ that $ G = HK $.then we can let $ K = K_{\pi}K_{\pi^{\prime}}$, that ...
1
vote
2answers
49 views

Sylow Theorems to find all groups with order less than or equal to 10

I'm trying to solve the following problem using the Sylow theorems: Determine all groups of order $\leq 10$ up to isomorphism. I know that in particular I have to use the fact that any ...
5
votes
2answers
75 views

$S_6$ contains two subgroups that are isomorphic to $S_5$ but are not conjugate to each other

This is a problem from Ph.D. Qualifying Exams. Show that the symmetric group $S_6$ contains two subgroups that are isomorphic to $S_5$ but are not conjugate to each other. Here is my method. $S_5$ ...
2
votes
2answers
35 views

Homomorphism with intersection of all Sylow p-subgroups as kernel?

Does anyone know of a homomorphism from a group $G$ to another group with kernel as the intersection of all Sylow $p$-subgroups? I was trying to prove that the intersection of Sylow subgroups is ...
2
votes
0answers
37 views

The group $G$ has order 56. Show that it contains normal Sylow p-subgroup.

I'm new at this topic, so I'm not sure about whether is my solution acceptable or not. Could you check it? By Sylow theorem there exist $|P|=7$ and $|Q|=8$. Sylow $p$-subgroup is normal iff it is ...
1
vote
2answers
37 views

How many $2$-Sylow subgroups in $G$ with $|G| = 2^2\cdot 3$?

I have a group $G$ with $|G| = 2^2\cdot 3$. I also know it has $4$ Sylow-$3$ subgroups. I need to show that there is $1$ Sylow-$2$ subgroup. (This is all I have left from the full question.) Any ...
5
votes
1answer
115 views

Every group of order $150$ has a normal subgroup of order $25$

Let $G$ be a group of order $150$. I must show that it has a normal subgroup of order $25$. The hint says to show that is has a normal subgroup of order $5$ or $25$. Now from Sylow, I know that the ...
4
votes
2answers
55 views

Showing Sylow $p$-groups are Abelian if $n_p=2p+1$

An old qual question on Sylow $p$-groups: Assume that $p$ is an odd prime and $G$ is a finite simple group with exactly $2p+1$ Sylow $p$-groups. Prove that the Sylow $p$-groups of $G$ are abelian. ...
0
votes
1answer
30 views

Sylow Theorems Problem..

Suppose $G$ is a simple group of order $315=3^2\times 5\times 7.$ Then I can prove that $G$ has $84$ elements of order $5$ and $90$ elements of order $7.$ But when we consider the Sylow $3$ ...
1
vote
1answer
47 views

group $G$ of order $312$. show that G is not simple

I have a group G of order $312$ and I need to show $G$ is not simple. What I tried : I know $312 = 2^3\times39$ so, I know that I have an element of order $2$. does that mean I have a subgroup ...
0
votes
1answer
64 views

Number of elements of order $11$ in group of order $1331$

Let $G$ be a group of order $1331$. Prove that $G$ has at least $11$ elements of order $11$. $|G|=1331=11^3$ So by First Sylow's theorem, there exists a Sylow $11$-subgroup of G. By Third Sylow's ...
1
vote
1answer
46 views

$G$ finite, $P < G$ a Sylow p-subgroup, $N_{G}(P)$ the normalizer is contained in $H < G$, show $N_{G}(H)=H$

Let $G$ be a finite group and $P<G$ be a Sylow p-subgroup. Let $N_{G}(P)$ be the normalizer of $P$ in $G$. Let $H<G$ be a subgroup containing $N_{G}(P)$. Prove that $N_{G}(H)=H$. I've been ...
1
vote
0answers
28 views

intersection of all $2$-sylow in $S_n$

I am currently stuck on an exercise that asks to determine explicitly the subgroup $O_2(S_n)<S_n$ given by the intersection of all Sylow $2$-subgroups, for $n\geq 4$. I already proved that ...
2
votes
1answer
40 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
27 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
83 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
59 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
119 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 ...
5
votes
2answers
65 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 ...
15
votes
4answers
540 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
73 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
90 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
67 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
61 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
79 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
67 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 ...
-1
votes
1answer
52 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
36 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
66 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
79 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
28 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
157 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
56 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
129 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
123 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 ...