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

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Intersection of all $p$-Sylow subgroups is normal

Let $G$ be a finite group, $p$ a prime number that divides $|G|$ and $O_p(G)=\bigcap_{P \in Syl_p(G)}P$. Prove that 1) $O_p(G) \lhd G$ 2) $O_p(G)$ is maximal among the normal $p$-subgroups of $G$. ...
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Which are the nine Sylow $2$-subgroups of $S_3 \times S_3$? What is the only Sylow $3$-subgroup of $S_3 \times S_3$? And the most important…Why? [on hold]

Which are the nine Sylow 2-subgroups of $S_3 \times S_3$? What is the only Sylow 3-subgroup of $S_3 \times S_3$? And the most important... why? I am doing an independent study of Abstract Algebra. ...
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For any group $G$, $|G/Z(G)| \neq 91$.

In Malik's Fundamentals of abstract algebra, one can find the following problem: Prove that for any group $G$, $\vert G/Z(G)\vert \neq 91$. This exercise is just ahead of Sylow's theorems. I've ...
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Groups of order 24.

I supposed $n_3=4$ and $n_2=3$, and then I made $G$ act by conjugation on $Syl_3 (G)$. I want to show that $G\cong S_4$ (looking at all order 24 groups here ...
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Sylow $p$-subgroup of $GL_{n}(\mathbb{F}_{q})$

This question has already been asked here. However, the answer there isn't particularly helpful. Consider $G = GL_{n}(\mathbb{F}_{q})$ where $q = p^{r}$ where $p$ is prime. Show that the upper ...
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Groups of order 36, normalizer and left multiplication.

I proved using left multiplication action that Every group of order 36 has a subgroup of order 9 (and 18), if $n_3=1$, and a subgroup of order 3 if $n_3=4$. Using normalizers I could prove that Every ...
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71 views

Why is $G$ abelian?

If $|G|=pq^2$ with $p,q$ primes and if $p<q$, with $q\not\equiv\pm1\mod p$, why is $G$ abelian ? The $3^{rd}$ Sylow theorem implies that $n_p|q^2$ and $n_p\equiv 1 \mod p$, By hypothesis, ...
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Isomorphism type of two quotient groups of $G=\mathbb{Z}^\times_{16}$

I'm supposed to determine the isomorphism type of $G/\langle 15\rangle $ and $G/\langle 9 \rangle$. I've determined the order of both subgroups ($\langle 15\rangle$ and $\langle 9\rangle$), and it is ...
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Show that a group of order $180$ is not simple.

What i deduced is that $n_5=1,6$ or $36$. We are done if $n_5=1$. If $n_5=36$ we $N_G(P)=P$ for any Sylow $5$-subgroup P as $|N_G(P)|=\frac{180}{36}=5$ and $P$ is abelian cyclic so by Burnside ...
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Every finite $p$-group is solvable

I know that in some version of Sylow's 1st theorem, it states that if $|G|=p^nm$ for some $n\geq 1$ and where $p$, a prime number, does not divide $m$, then every subgroup $H$ of $G$ of order ...
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1answer
37 views

Every $p$-subgroup is contained in one $p$-Sylow subgroup?

I am learning Sylow's theorems in my algebra course and I was reading questions posted before. One is the following: If $H$ is a normal subgroup of a finite group $G$ and $|H|=p^k$ for some prime $p$. ...
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42 views

T/F questions on $p$-Sylow subgroups and normalizer

Prove the following statement if it is true. Otherwise, disprove it by giving a counterexample. 1) The normalizer $N$ in a finite group $G$ of a subgroup $H$ of $G$ is always a normal subgroup ...
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38 views

Semidirect Product Definition of Addition

Hello Everyone, I'm having a hard time trying to define the addition on this semidirect product, any suggestion would be appreciated. Thanks.
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Number of Sylow 2-subgroups of $D_{2n}$

I want to solve the following exercise from Dummit & Foote's Abstract Algebra text: Let $n=2^ak$ where $k$ is odd. Prove that the number of Sylow 2-subgroups of $D_{2n}$ is $k$. [Prove that if ...
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37 views

Density of Sylow subgroups

Let G be any group of order n.Also assume p be the largest prime dividing n.Let n(p) be the maximum no of sylow subgroups a group of order n can have.Is it possible to sa anything definite about the ...
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53 views

Group of order $18$ contains exactly one subgroup of order $9$

I'm trying to prove the following: Proposition: A finite group $G$ of order $18$ has a unique subgroup of order $9$. Here is my attempt: Observe that $18 = 3^2 \times 2$. Let's count the number ...
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55 views

Simple group with Klein four Sylow

If $G$ is a simple group, with a Sylow $2$-subgroup isomorphic to the Klein four group $\mathbb{Z}_2 \times \mathbb{Z}_2$, then I want to show that any two involutions in a given Sylow $2$-subgroup ...
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38 views

Finding Sylow p-subgroups

I am trying some examples of finding Sylow p-subgroups in specific groups and looking for the most efficient way to do so. For example, lets say we need to find Sylow-3 subgroups in $A_4$ and $D_6$, ...
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Is it true that $n_p!\le |G|$?

Let $G$ be a finite group and $n_p:=|\text{Syl}_p(G)|$. Is it true that $n_p!\le |G|$ ? I've shown that it's true, but I'm not so sure, can you check my proof? Proof. Let $G$ act on ...
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$2$-groups with odd permutations

If $P$ is the Sylow $2$-subgroup of a finite group $G$, $H <P$, and $x \in P$ so that no non-trivial element of $\langle x \rangle$ conjugates into $H$ (in $G$), and $|P|=|H||x|$, how can I show ...
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Question about Sylow subgroups

"Let $|G|=p_1^{e_1}\cdot \cdot \cdot p_t^{e_t}$ and let $G_{p_i}$ be the Sylow $p_i$-subgroup of $G$. The subgroup $S$ generated by all of the Sylow subgroups is $G$, for $p_i^{t_i}|~|S|$ for all ...
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Groups of order $p(p+1)$

If I have a group of order $p(p+1)$ with $p+1$ Sylow $p$-subgroups how can I prove that all $p$ non-trivial elements not of order $p$ have prime order?
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When do Sylow $p$ and Sylow $q$ subgroups commute?

Do $p$-Sylow and $q$-Sylow subgroups commute iff both are unique and thus normal? I know that one direction is true: namely that if the $p$-Sylow subgroup and the $q$-Sylow subgroup are normal in the ...
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Group with $p+1$ Sylow $p$-subgroups

Given a group $G$ with $p+1$ Sylow $p$-subgroups, I've deduced that $R = P \cap P'$, where $P, P'$ are Sylow $p$-subgroups, has index $p$ in each of $P, P'$; and that all $p+1$ Sylow $p$-subgroups of ...
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56 views

Group of order 36

I follow a proof which show that any group of order $36$ has a normal Sylow subgroup. The author suppose that $G$ has no normal subgroups of order $9$ or $4$. Then $G$ won't have subgroups of order ...
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1answer
41 views

Normalizer and centralizer are equivalent when $p$ is the smallest prime dividing $|G|$

Let $p$ be the smallest prime dividing $|G|$, and suppose that some $P \in \mathsf{Syl}_p(G)$ is cyclic. Prove that $N_G(P) = C_G(P)$. So I let $G=p^\alpha m$ $p$ does not divide $m$. P is cyclic, ...
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Applying Sylow Theorems to a group of order 24.

Prove that a group $G$ of order 24 with no element of order 6 is isomorphic to the symmetric group $S_4$. Here is my approach. I believe this is an alternative proof to the one provided here: Group ...
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1answer
53 views

Prove that a non-abelian group of order $pq$ ($p<q$) has a nonnormal subgroup of index $q$

So I've come up with a proof for the following question, and I'd like to know if it's correct (as I couldn't find anything online along the lines of what I did). Question Let $p$ and $q$ be primes ...
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Sylow's Theorem Explanation

Can someone explain it to me? I've been working out of Galian's Contemporary Abstract Algebra this semester, but came into possession a copy of Dummit and Foote's book, which I am aware is ...
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1answer
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Hall's Subgroup Theorem

I am currently looking into the extension of Sylow's theorems, namely through Hall-$\pi$-subgroups and Hall's Theorem. I currently have the theorem as; Let $G$ be a finite solvable group a $\pi$ be ...
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Lang Sylow Theorem

STATEMENT: Let $P$ be a $p$-Sylow subgroup of $G$ and H is a $p$-subgroup of G. Suppose first that $H$ is contained in the normalizer of $P$. We prove that $H\subseteq P$. Indeed, $HP$ is then a ...
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If $G$ is a group of order $48$, show that the intersection of any two distinct Sylow $2$-subgroups has order $8$

All I know is that we have $3$ Sylow-$2$ subgroups of order $16$. $$o(H \cap K)= o(H)o(K)/o(HK)$$ How to proceed further?
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Prove that $n_p(N)$ divides $n_p(G)$

Let $N$ be a normal subgroup of $G$ where $G$ is finite group, then we have to prove $n_p(N)$ divides $n_p(G)$ ( here $n_p(G)$ means number of sylow $p$-subgroups of $G$) I was able to prove that ...
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30 views

Group of order $|G|=pqr$, $p,q,r$ primes has a normal subgroup of order

Let $p,q,r$ be positive primes, $p<q<r$, and let $G$ be a group with $|G|=pqr$. Show that there exists a normal subgroup $H$ of $G$ of order $qr$. I've seen this post Groups of order $pqr$ and ...
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42 views

Sylow subgroups of a non abelian group $G$ with $|G|=21$ and $|G|=39$

I am trying to solve the following exercise: ¿How many Sylow subgroups has a non abelian group $G$ of order $21$ and $39$ respectively. I could do the following: a) $|G|=21=3\cdot 7$. I'll call ...
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49 views

There are no simple groups of order $27p$

Statement Show that there are no simple groups of order $27p$ for any $p$ prime number. I got stuck with this problem, I'll write what I've done so far: Suppose $G$ is a group with $|G|=3^3p$, $p$ ...
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Sylow subgroups problem

Let $G$ be a finite group and $p<q$ such that $p^2$ doesn't divide $|G|$. Let $H_p$ and $H_q$ be Sylow subgroups of $G$ with $H_p \lhd G$. Show $H_pH_q \lhd G \space \implies H_q \lhd G$. From the ...
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1answer
45 views

Sylow p-subgroups and Sylow theorem

Find all Sylow 3-subgroups of $S_3\times S_3$? This is what I already found: Since $O(S_3\times S_3)=36=2^2 3^2$ Sylow- $3$ subgroups have order $9$. If $n_3$ is the no. of Sylow- $3$ subgroups, Then ...
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A group $G$ with $|G|=p^3q$ and with no normal Sylow subgroups is $G \cong \mathbb S_4$

Problem Show that if $|G|=p^3q$ and $G$ has no normal Sylow subgroups, then $G \cong \mathbb S_4$ The attempt at a solution By the Sylow theorems we have: -$n_p \equiv 1 (p), \space n_p|q$ -$n_q ...
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A p subgroup of a group is contained in a sylow p subgroup

I am trying to solve a problem involving automorphism of a group.There needs the following argument: a p subgroup is contained in a sylow p subgroup.Is it true?I can't prove it,may be it is ...
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Number of Sylow 2-subgroups of a special linear group

Find the number of Sylow $2$-subgroups of the special linear group of order 2 on $\mathbb{Z}$ (modulo $3$). I think it will be $1$. But I failed to prove it using the counting principle. It has $4$ ...
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51 views

If $G$ is a finite group whose $p$-Sylow subgroup $P$ lies in its center, then there is a normal subgroup $N$ of $G$ with $P\cap N=\{e\}$ and $PN=G$

If $G$ is a finite group and its $p$-sylow subgroup $P$ lies in the center of $G$, prove that there exists a normal subgroup $N$ of $G$ with $P\cap N=\{e\}$ and $PN=G$
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45 views

Sylow Subgroup is

G be a finite group.P is a Sylow p-subgroup which is contained in the center of G.Show that there is a normal subgroup N of G such that G=PN.(Herstein problem,page 103 prob 16) Give some idea.P is ...
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Two Open Ended Questions in Sylow Theory

Sylow Theorems are very powerful in finite group theory. Two natural questions come to mind: 1) Given a finite group $G$ and a $p$-subgroup $H$ of $G$, how many Sylow $p$-subgroups of $G$ contain ...
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A Group Having a Cyclic Sylow 2-Subgroup Has a Normal Subgroup.

I want to prove the following: Let $G$ be a group of order $2^nm$, where $m$ is odd, having a cyclic Sylow $2$-subgroup. Then $G$ has a normal subgroup of order $m$. ATTEMPT: We ...
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127 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 ...
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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 : ...
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54 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} ...
2
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4answers
83 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
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1answer
58 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 ...