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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1answer
34 views

Abstract Algebra: Proof involving a p-Sylow, nomal subgroup P in G. [closed]

Let $P$ be a normal subgroup in $G$ where $P$ is a $p$-Sylow subgroup of $G$. Show that $\phi(P)=P$ for every automorphism $\phi$ of $G$.
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2answers
50 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 ...
8
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2answers
183 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
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2answers
59 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
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2answers
74 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: ...
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2answers
63 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
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3answers
52 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
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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
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1answer
69 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
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2answers
62 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 ...
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1answer
49 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 $q?$
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2answers
24 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
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1answer
59 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
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1answer
72 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 ...
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0answers
26 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 ...
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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 ...
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3answers
119 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 ...
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1answer
39 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 ...
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1answer
125 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 ...
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1answer
121 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 ...
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1answer
23 views

Sylow $p$-subgroup of a normal group.

Let $G$ be a transitive subgroup of $S_p$ and let $H$ be a non-trivial normal subgroup of $G$. I need to show that any Sylow p-subgroup of $G$ is also contained in $H$. I know that any transitive ...
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2answers
23 views

Sylow $p$-group of a transitive subgroup of $S_p$

How do I show that any transitive subgroup of $S_p$ contains a non-trivial Sylow $p$ subgroup, of cardinality $p$? I am trying to prove a result of Galois and the only hint I have is that if $p$ is a ...
3
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1answer
69 views

The number of Sylow subgroups on $G$ with $|G|=pqr$

I'm doing a part of an exercise and I don't know how to go on. Here it goes: Let $G$ be a group, with $|G|=pqr$, $p,q,r$ different primes, $q<r$, $r \not\equiv 1$ (mod $q$), $qr<p$. Show $G$ ...
2
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1answer
40 views

Let $G$ be a group, with $|G|=pqr$, $p,q,r$ different primes, $q<r$, $r \not\equiv 1$ (mod $q$), $qr<p$. Show $G$ has an unique Sylow $p$-subgroup.

I'm stuck on this exercise: Let $G$ be a group, with $|G|=pqr$, $p,q,r$ different primes, $q<r$, $r \not\equiv 1$ (mod $q$), $qr<p$. Show $G$ has an unique Sylow $p$-subgroup $P$. What ...
1
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2answers
77 views

Let $G$ be a finite group and $H < G$. Prove that $n_p(H) \le n_p(G)$.

Let $G$ be a finite group and $H < G$. Prove that $n_p(H) \le n_p(G)$. Ok, so now i know that $n_p(H) \le n_p(G)$ refers to the number of Sylow p-subgroups in H and G, respectively. From here, I ...
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1answer
38 views

Find the number of Sylow $p$-subgroups of $G$, if we know that $\lvert G\rvert=6$

Today I've been looking the third Sylow theorem. My professor did an example in class, so I tried to solve the example by myself and then compare what I did with the answer of my professor. The ...
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1answer
33 views

Why is the group abelian?

Lets say a group $G$ consists of 3 Sylow groups, $H_1,H_2,H_3$. Each of order $p_1,p_2,p_3$, that are prime numbers and different. Since we only have one of each Sylow group for each p, the second ...
1
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1answer
57 views

Let $P$ be a Sylow p-subgroup of a finite group $G$ and let $H$ be a normal subgroup of $G$. Show if $p$ does not divide $[G:H]$, then $P \subseteq H$

Let $P$ be a Sylow p-subgroup of a finite group $G$ and let $H$ be a normal subgroup of $G$. Show that if $p$ does not divide $[G:H]$, then $P \subseteq H$ I do not understand how to work this ...
2
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1answer
100 views

About first Sylow Theorem proof

I've been struggling with the first Sylow theorem proof we were given in class. This is how my professor introduced us the theorem: First Sylow Theorem: Let $G$ be a finite group. Let $\rvert G ...
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2answers
59 views

Let G be a group of order $p^2q^2$ where $p > q$ are primes. If $|G| \ne 36$, show that G has exactly one Sylow p-subgroup

Let G be a group of order $p^2q^2$ where $p > q$ are primes. If $|G| \ne 36$, show that G has exactly one Sylow p-subgroup. Ok, I'm not exatly sure where to begin with this one. I know that we are ...
2
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2answers
48 views

Let $S$ be a normal p-subgroup of a finite group $G$. Prove that $S \subseteq P$ for every Sylow p-subgroup $P$ of $G$

Let $S$ be a normal p-subgroup of a finite group $G$. Prove that $S \subseteq P$ for every Sylow p-subgroup $P$ of $G$. Now, I know that this involves the Sylow Theorems, of course. This is very new ...
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1answer
66 views

Exercise 257 from Rose“ A course on group theory”

I would please like some help on the exercise 257 of the Rose Group Theory. Let $G$ be a finite group and $H$ and $K$ be normal subgroups of $G$, and $P$ a Sylow p-subgroup of $G$. Then $(PH)\cap ...
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1answer
45 views

Let $G$ a finite group such that $\lvert G \rvert=pm$, with $p$ a prime and $\gcd(p,m)=1$. $G$ has an unique Sylow $p$-subgroup $P$. Prove $P\lhd G$.

I just made this exercise, left as homework, and I'm almost sure that I did something wrong, or at least that there's a better way to solve it. Here it goes: Let $G$ a finite group such that ...
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2answers
44 views

How to use Sylow theorems to prove that a subgroup of $SL_2(\mathbb{F}_3)$ is normal.

I have $G = SL_2(\mathbb{F}_3)$ and $H = \langle i, j \rangle$ where $$ i = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}, \quad j = \begin{pmatrix} -1 & 1 \\ 1 & 1\end{pmatrix} $$ (In ...
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3answers
44 views

Does $P \in Syl_p(N_G(P))$ implies $P \in Syl_p(G)$?

In a finite group $G$, let $P \in Syl_p(N_G(P))$, i.e. $P$ is a Sylow $p$-subgroup in its normaliser, does this imply $P \in Syl_p(G)$, i.e. it is a Sylow $p$-subgroup in its entire group?
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38 views

Sylow Subgroups of a Dihedral Group

Let $G = D_{2n}$ and $p$ an odd prime dividing $2n$, i.e. $2n = 2m\cdot p^{\alpha}$ with $p \nmid 2m$. I need to show that $P \in \operatorname{Syl}_p(G)$ is normal in $G$ and cyclic. As far as ...
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3answers
67 views

A group of order $1645$ is cyclic

This is what I have so far, but I get stuck at one point. $|G| = 1645 = 5\cdot 7\cdot 47$ I found with the Sylow theorem that there is a unique Sylow $5$-Subgroup, a unique Sylow $7$-Subgroup and ...
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1answer
52 views

Sylow theory question

I am asked: Does a group $G$ of order 14 have to be cyclic? Using Sylow's third theorem (work omitted), I conclude that $$|\textrm{Syl}_2(G)|=1 \textrm{ or }7$$ and $$|\textrm{Syl}_7(G)|=1$$ I ...
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1answer
63 views

A finite group of order $pq$ cannot be simple.

Let $p$ and $q$ be prime numbers. I wish to prove that a finite group $G$ of order $pq$ cannot be simple. Proof. Case 1: $p\not= q$. Case 2: $p=q$. Consider the first case where $p\not= q$. Without ...
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1answer
31 views

Groups and Characters exercise hint

I am trying to make my way through Grove's Groups and Characters but am having some trouble with the following seemingly benign exercise: If a group $G$ has a normal $p$-complement $K$ show that ...
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2answers
47 views

How do they know that the Sylow 3 subgroups are cyclic?

I have this exercise: Find all Sylow 3 subgroups of $S_4$ and demonstrate that they are all conjugate. This is the answer given in the manual: Now, Sylows first theorem says that every Sylow 3 ...
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0answers
45 views

exercise Sylow theorem

I have an exercise regarding Sylow theorems. I am supposed to add a number where it is blank. A Sylow 3-subgroup of a group of order 12 has order...?(the answer in the appendix is given to be 3) ...
3
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1answer
48 views

Sylow subgroup of a group

Let $G$ be a group such that $\vert G\vert=231$. I have to show that the unique Sylow 11-subgroup of $G$ is contained in the center of $G$ I proceed as follows: Since the number of Sylow 11-subgroup ...
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2answers
30 views

Why $n_i=j$ implies that there is a subgroup $H$ of $G$ of index j?

Why $n_i=j$ implies that there is a subgroup $H$ of $G$ of index j ? If I denote with $n_i$ the number of i-Sylow subgroups For example if $|G|=180=2^2\cdot3^2\cdot 5$ then if I assume that ...
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1answer
71 views

Group of order 396 isn't simple

Prove that group of order $396=11\cdot2^2\cdot3^2$ is not simple. $n_{11}$ is $1$ or $12$, so I assumed $n_{11}=12$ and tried to look at the action of the group on $Syl_{11}\left(G\right)$ by ...
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1answer
72 views

Prove any group of order $185$ is cyclic.

This is my attempt. I am not sure as for its plausibility. $Attempt$: Let $G$ be a group of order $185$. Then $G=185=5\cdot 37$. The $Sylow-p$ subgroups are unique and normal and therefore $G$ is ...
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1answer
36 views

Verify my proof of $G$ is nilpotent iff $xy=yx \forall x,y\in G$ such that $(o(x),o(y))=1$.

Prove: $G$ is nilpotent iff $xy=yx \forall x,y\in G$ such that $(o(x),o(y))=1$ $G$ is finite. Is that plausible? Attempt: Suppose $G$ is nilpotnet. Then $G=P_1\times\ldots\times P_k$ where ...
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1answer
116 views

Non Abelian group of order pq

Given primes p & q with q dividing $p-1$, construct a non-abelian group of order pq as follows: Let P have order p and let Q $\subseteq$ Aut(P) have order q. Let G $\subseteq$ Sym(P) be the set of ...
4
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1answer
62 views

$G$, $|G|=n$ is nilpotent $\iff$ $\forall m|n$, $G$ has a normal subgroup of order $m$.

Prove $G$, $|G|=n$ is nilpotent $\iff$ $\forall m|n$, $G$ has a normal subgroup of order $m$. I got stuck in the second direction. One direction: $|G|=n=p_1^{s_1}\cdot ...\cdot p_k^{s_k}$ Where $p_i$ ...
0
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1answer
40 views

Image of Sylow Subgroup under Automorphism

Let G be finite with G $>$ 1 and suppose P $\subseteq$ Aut(G) is a p-subgroup. Show that there exists some nontrivial Sylow q-subgroup Q of G (for some prime q) such that $\sigma$(Q)=Q, $\forall$ ...