Questions tagged [sylow-theory]

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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Groups and Cosets

I am studying the proof of the third Sylow's Theorem and I dont get this: Let $G$ be a finite group, $H$ a $p$-Sylow subgroup of $G$, $N(H)= \{g \in G : gH=Hg \}$. Note that $G = \cup_{g \in G}AgA$ (...
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Sylow 2-subgroup of Suzuki Group $Sz(8)$

I need to find the isomorphism class containing the Sylow 2-subgroup of the Suzuki group $Sz(8)$. Can anyone give a reference?
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Is every infinite group nilpotent iff it is direct product of its sylow p-subgroups?

We know that every finite group is nilpotent iff it is direct product of its sylow p-subgroups.$$$$is this also true for infinite groups?
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$S$-semipermutable subgroups

A subgroup $H$ of a finite group $G$ is said to be $S$-semipermutable in $G$ if it permutes with every Sylow $q$-subgroup of $G$ for all primes $q$ not dividing $|H|$. Assume that $G$ is solvable and ...
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Normalizers of Sylow subgroups.

If $G\neq \left \{ 1 \right \}$ is a finite solvable group, then there is at most one prime $p$ such that if $P\in \operatorname{Syl}_p(G)$, then $N_G\left ( P \right )=P.$ I think it's necessary to ...
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1 vote
1 answer
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When should Sylow subgroups intersect and when they should not?

Here is the question I am trying to understand its solution: Prove that a group of order $11 \times 2^{10}$ has a normal subgroup. And here is a solution I found to the part of excluding the case $n_2 ...
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Are Sylow $p$-subgroups in infinite groups conjugate? [closed]

In a finite group $G$, if $P$ is a Sylow $p$-subgroup of $G$ then every Sylow $p$-subgroup is conjugate to $P$, but when $G$ is an infinite group, is this still true? I was suggested to use the ...
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-3 votes
1 answer
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Excluding the case $n_2 = 11$ and $n_{11} = 1024.$ [duplicate]

Here is the question I am trying to solve: Prove that a group of order $11 \times 2^{10}$ has a normal subgroup. And here is a solution I found to the part of excluding the case $n_2 = 11$ and $n_{11} ...
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2 votes
1 answer
148 views

Prove that a group of order $11 \times 2^{10}$ has a normal subgroup.

I am trying to solve this question: Prove that a group of order $11 \times 2^{10}$ has a nontrivial proper normal subgroup. My trial By Sylow theorems I know that $n_2 \in \{1,11\}$ and $n_{11} \in \{...
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Show that $SL_2(F_3)/Z(SL_2(F_3)) \cong A_4$

Show that $SL_2(F_3)/Z(SL_2(F_3)) \cong A_4$ I know that $|SL_2(F_3)/Z(SL_2(F_3))|= 12$. Then if the quotient group has a normal subgroup of order $4$ then it is isomorphic to $A_4$. Suppose that it ...
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Prove that groups of order 200 are solvable

Suppose $G$ is a group with $|G| = 200$. Since $200 = 2^3 \cdot 5^2$ I've used Sylow's theorem to make two claims concerning $n_G(8)$, the number of Sylow 2-subgroups of $G$: $$ n_G(8) \equiv 1 \mod ...
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Abelian group of order $pq$ is a subgroup of $S_{p+q}$

Let $G$ be a finite abelian group of order $pq$, where $p<q$ are both primes. I want to show that $G$ is isomorphic to a subgroup of $S_{p+q}$ (but is not isomorphic to any subgroup of order $S_{p+...
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5 votes
1 answer
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Quotient of the Galois group of a splitting field generated by two roots

Let $f \in \mathbb{Q}[x]$ be irreducible of degree $p$, where $p$ is a prime. Let $K$ be the splitting field of $f$ and suppose that there are roots $\alpha$ and $\beta$ of $f$ such that $K = \mathbb{...
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Action of the normalizer of the complement

Let $G$ be a finite group, and let $p$ be a prime. Let $U$ be a Sylow $p$-subgroup of $G$. By the Schur-Zassenhaus theorem, there is a complement $T$ to $U$ in $N_G(U)$. Observe that $N_G(T)$ acts by ...
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1 vote
1 answer
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Nonabelian group of order 28.

Is there a nonabelian group of order 28 whose 2-Sylow subgroup is isomorphic to $\mathbb{Z}/4\mathbb{Z}$? My reasoning is that by Sylow's Theorem, there is a 2-Sylow subgroup of order $4$. Since it is ...
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1 answer
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Intersection of all Sylow $p$-subgroups

I wanted to prove that the intersection of all Sylow $p$-subgroups of a finite group G is a normal subgroup of $G$. Can someone enlighten me how is this implication possible: If an automorphism $\...
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4 votes
1 answer
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Number of Sylow $p$-subgroup of $\mbox{SL}_n(\Bbb F_p)$.

What is the number of Sylow $p$-subgroup of $\mbox{SL}_n(\Bbb F_p)$ where $\Bbb F_p$ is finite field of order $p$? This problem is known for $\mbox{GL}_n(\Bbb F_p)$. By checking the order, strictly ...
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2 votes
2 answers
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If $G$ has no subgroup of index $2$ and $G\leq S_n$, then $G \leq A_n$.

I am currently reading Abstract Algebra by Dummit & Foote. Discussing some techniques about the Sylow theorems they prove the proposition mentioned in the title (Proposition 12 (1), p. 204, 3rd ...
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$2$-Sylowgroup of $A_6$

I want to find a $2$-Sylowgroup, i.e. a subgroup of order $8$, of $A_6$. I think I can solve this problem by brute force but in order to avoid endless calculation I was wondering whether someone could ...
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7 votes
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Proof there are no perfect groups of order 3024

How can I prove that there are no perfect groups of order $3024$? My attempt is the following: Each non-trivial finite perfect group admits a non-abelian simple quotient. This holds because if the ...
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1 vote
0 answers
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No group of order 160 is simple [duplicate]

I'm trying to prove that no group of order 160 is simple. The following is my approach. Let $G$ be a group of order $160$. Note $160 = 2^55$. I can easily get that $n_2 = 1$ or $5$ and $n_5 = 1$ or $...
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1 vote
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Let $p$ be prime. Let $S$ be a Sylow $p$-subgroup of $G$. If $x\in G$ then $xSx^{-1}$ is a Sylow $p$-group. [duplicate]

Let $p$ be prime. Let $S$ be a Sylow $p$-subgroup of $G$. If $x\in G$ then $xSx^{-1}$ is a Sylow $p$-subgroup. My attempt: Let $|G|=p^n\cdot m$ where $p$ is prime, $n,m\in\mathbb{Z}$ such that $p\nmid ...
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2 answers
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Why are Sylow Theorems and Sylow subgroups significant?

If one read's Gallian's Abstract Algebra book then they would find that the chapter for Sylow Theorem's is quite hyped up. However, I am unable to understand the big picture of why Sylow subgroups and ...
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1 answer
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Do isomorphic groups have the same number of Sylow $p$-subgroups?

I think that isomorphic groups should have the same number of Sylow $p$-groups, but I am not sure why, I am a little stuck on this, I really don't know where to even begin, or if this is even true (...
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15 votes
2 answers
281 views

Can we construct a group with exactly $k$ Sylow-Subgroups?

Inspired by the answers given by these three questions (here, here, and here), what is the general solution for constructing a group with a specific number of Sylow subgroups? That is, given a prime $...
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Alternate proof of Sylow's First theorem.

I am trying to prove first Sylow Theorem using the Lemma: if $G$ is a finite group such that has a Sylow $p$-subgroup and $H\subset G$, then $H$ has a Sylow $p$-subgroup. The way I want to go about ...
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Finite groups $G$ in which maximal subgroups of Sylow subgroups are normal in $G$

If $G$ is nilpotent, then maximal subgroups of Sylow subgroups of $G$ are all normal in $G$. So the class of groups those finite groups with property that maximal subgroups of Sylow subgroups are ...
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-1 votes
1 answer
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$G$ is supersolvable if the maximal subgroups of Sylow subgroups of $G$ are normal in $G$

$G$ is supersolvable if the maximal subgroups of its Sylow subgroups are normal in $G$. Then $G'$ is nilpotent. If $P$ is a non-normal Sylow subgroup of $G$, then why is $P$ not contained in $G'$?
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2 votes
1 answer
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Strategy for classifying some groups of order $pqr$ - recognizing direct factors

I've been reviewing some of my notes from an abstract algebra class that I took and have been thinking about/redoing some of the examples we did classifying groups of smallish order $pqr$. In ...
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3 votes
1 answer
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non-abelian groups of order of product of 3 distinct primes

Suppose that $p,q,r$ are distinct primes such that $q = 1 \pmod p$ $pq = 1 \pmod r$ Consider the following group $G$ of order $pqr$: $1$ element must be the identity element Exactly $q-1$ elements are ...
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0 votes
1 answer
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What is the question "If G is group of order 360 what are the possible isomorphism types of the Sylow p-subgroups for p=2,3,5,7?" really asking?

What is the question "If G is group of order 360 what are the possible isomorphism types of the Sylow p-subgroups for p=2,3,5,7?" really asking? I am familiar with the Sylow theorems, but ...
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1 vote
1 answer
54 views

Number of subgroups of given order is a product of prime powers congruent to $1\pmod p$

If we have a group $G$, and maximal a prime power $p^k$ divides $|G|$ (meaning that $p^{k+1}$ does not divide $|G|$), then we must have a subgroup $H$ of order $p^k$, by Sylow's first theorem. Let $...
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1 answer
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Let $G$ be a group of order $2014$. Prove that $G$ has a normal subgroup of order $19$ and $G$ is solvable [duplicate]

Let $G$ be a group of order $2014$. Prove that $G$ has a normal subgroup of order $19$ and $G$ is solvable. The first part directly follows from the Sylow Theorems, if you write $2014 = 2 \cdot 19 \...
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1 answer
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When is a Sylow $p$-subgroup normal?

Let $A_5$ be the alternating group of degree 5. I would like to prove that the number $s_5$ of Sylow 5-subgroups of $A_5$ ist 6. With $|A_5| = \frac{5!}{2} = 60 = 5 \cdot 12$ and the Sylow theorems I ...
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3 votes
0 answers
59 views

Exponent of $p$-sylow subgroup of symplectic groups over the field $GF(p)$

Let $P$ be the $p$-sylow subgroup of symplectic group $Sp(2n,p)$, where $p$ is an odd prime. I want to know if there exists anything about the exponent of $P$? I know that $Sp(m, p) \leq GL(m, p)$ ...
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Relation between $p$-Sylow subgroups of a group $G$ and the $p$-Sylow subgroups of $G/H$ [duplicate]

I'm learning about the Sylow theorem and the p-Sylow subgroups. In certain applications of the Sylow Theorem one considers the $p$-Sylow subgroups of the quotient group $G/H$ to conclude a statement ...
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1 answer
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Why are these two Sylow subgroups disjoint?

I have the following problem: Let $G$ be a finite group with $card(G)=p^2q$ with $p<q$ two prime numbers. We denote $s_q$ the number of $q$-Sylow subgroups of $G$ and similarly for $p$. I have ...
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1 vote
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Contradiction in the application of the Sylow Theorem on the alternating group $A_5$

Somehow I get a contradiction in the following application of the Sylow Theorem. If $G$ is a group with cardinality $60 = 2^2 \cdot 3 \cdot 5$ there are exactly $4$ or $24$ elements of order $5$. I ...
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2 votes
1 answer
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There is no simple group of order $36$.

I tried to do this as an exercise and wanted to ask if my proof is correct or if it is missing something. Thank you so much. Let $G$ be a group such that $\lvert G \rvert = 36 = 2^2 \cdot 3^2.$ Show ...
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1 vote
0 answers
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Groups of order $16$ and their subgroups of index $2$

Question: For each of the groups $P$ of order 16 determine the classes of subgroups of index $2$ under the action of $\operatorname{Aut}(P)$. Context: For almost $60$ years it has seemed to me that on ...
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1 vote
1 answer
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If the Sylow-$2$-subgroup of a group $G$ is cyclic then can it surject $\mathbb{Z}_2\oplus\mathbb{Z}_2$

Suppose $G$ is a group such that Sylow-$2$-subgroup is cyclic. Then can it surject into $\mathbb{Z}_2\oplus\mathbb{Z}_2$? If $G$ is an abelian group and Sylow-2-subgroup is Cyclic, then it can not, by ...
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4 votes
1 answer
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Let $G$ be a group of order $7105$, show that the order of the center of $G$ is divisible by $35$.

Let $G$ be a group of order $5\cdot 7^2 \cdot 29$, show that the order of the center $Z(G)$ of $G$ is divisible by $35$. Now, $G$ contains only a $5$-sylow, $P_5$, and since the conjugacy action of $G/...
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2 votes
0 answers
18 views

Query regarding the order of an element in the centralizer.

I was reading a text where they describe the structure of non-abelian order $12$ groups, I quote a part of the text. "Let $G$ be a non-abelian group of order $12$. Let $P_3$ be a Sylow's $3$-...
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If $g: G \rightarrow S_4$ is a group homomorphism with $|\ker g|=24$, then the group $G$ is solvable.

$\DeclareMathOperator{\Ima}{Im}$ By the first isomorphism theorem of groups, we have that $$G/\ker g \cong g(G) \leq S_4 $$ Since the group $S_4$ is solvable, we know that every subroup of $S_4$ is ...
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0 answers
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How to find the element described in Sylow II

I'm currently doing this question: Find two Sylow $3$-subgroups $H_{1}$ and $H_{2}$ of the alternating group $A_{4}$ and an element $\sigma \in A_{4}$ such that $H_{2}= \sigma H_1\sigma^{−1}$. I can ...
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0 votes
0 answers
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A $2$-Sylow of $A_6$ is self normalizing

Let $A_6$ be the alternating group of order $6$. Let $R$ be a $2$-Sylow of $A_6$, I want to show that $R=N_{A_6}(R)$. What I have done is that we have that there is a monomorphism from $S_4$ to $A_6$ ...
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  • 185
3 votes
1 answer
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$G\le S_n$ in which all $g\ne e$ have $<\sqrt{n}$ cycles must be $\Bbb Z_p$ or $\Bbb Z_p\rtimes\Bbb Z_q$

Exercise 1.4.18 in Dixon & Mortimer's Permutation Groups is Let $G$ be a permutation group of degree $n$, and suppose that each $x\ne1$ in $G$ has at most $k$ cycles. If $n>k^2$, show that $G$ ...
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2 votes
1 answer
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Understanding the definition of Sylow $p$-subgroups

Here is the definition of Sylow $p$-group (source: wikipedia) For a prime number $p$, a Sylow $p$-subgroup of a group $G$ is a maximal $p$-subgroup of $G$, i.e. a subgroup of $G$ that is a $p$-group (...
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2 votes
1 answer
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Let $P\in{\rm Syl}_p(G)$ and $K$ a subgroup of $G$ containing $N_G(P)$. Show that $N_G(K)=K$

Question: Let $P\in{\rm Syl}_p(G)$ and $K$ a subgroup of $G$ containing $N_G(P)$. Show that $N_G(K)=K$ I'm wanting to use Frattini, but it looks like I need to use Frattini on $N_G(K)$. Certainly, $...
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2 votes
1 answer
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A question regarding the conditions imposed on an index in the course of the proof of Sylow's theorem I.

Source : Eliott Nicholson ( YT) , Group Theory Playlist, video n° 75, 22:40 [image below] Let $G$ be a group such that $|G|=p^{\alpha}m$ with $p$ a prime number and $m\in \mathbb N$. Let $\mathscr{C}$...
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