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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38 views

Prove the order of a group is congruent to $1 \pmod{p}$

Let $p$ be a prime number. Suppose $G$ is a group and $P$ is a Sylow $p$-subgroup. Define $N = \{g\in G\mid g^{-1}Pg = P\}$. Suppose $M\leqslant G$ and $N \subseteq M$. Prove that $|\{aM\mid a\in G\}|\...
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1answer
40 views

Cyclic Sylow 2-subgroup$\Rightarrow2$-nilpotent

In this, the author has shown that Burnside showed that if a group has a cyclic Sylow $2$-subgroup, then it is $2$-nilpotent. I try to read it and posted having problems with these lines in its proof. ...
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2answers
67 views

Number of elements of order $p$ in the sylow $p$-subgroup of $S_{p^2}$

I have the group $G = \mathbb{Z}_p \wr \mathbb{Z}_p$, it's well known that $G$ is isomorphic to a sylow $p$-subgroup of $S_{p^2}$, it has order $p^{p+1}$ and there are 2 possible orders for a non-...
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1answer
58 views

Understanding Sylow-Groups

I just read that a Group of order $12$ consists of Sylow-Groups of order $3$ and $2$. I also read that the number of elements of $Syl_2G$ is greater or equal than $4$. Why is that? Also, how can I ...
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1answer
51 views

Are the Sylow $p$-subgroups of $S_4$ also Sylow $p$-subgroups of $S_5$?

Here is my thinking: $|S_4| = 4! = 1 \times 2 \times 3 \times 4 = 2^3 \times 3$. $|S_5| = 5! = 1 \times 2 \times 3 \times 4 \times 5 = 2^3 \times 3 \times 5$. Since $2^3$ is the maximal power of $2$ ...
3
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1answer
68 views

Prove that if $|G| = 160$, $G$ is not simple.

I'm trying to prove this with Sylow's Theorem. I understand that the intersection between two Sylow-2 subgroups $H$ and $K$ cannot be of order $16$, since $| H \cap K| = 16$ implies $H \cap K \lhd G$. ...
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2answers
130 views

The center of a group of order $3^2q$ has order $1$ or $3$.

this is a question from a brazillian book, Paulo A. Martin's "Grupos, Corpos e Teoria de Galois" (portuguese for "Groups, fields and Galois Theory" (The actual word field translate ...
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2answers
75 views

Stuck in Proof of a Corallary to the Sylow Theorems

Corallary: If $G$ is a finite group with $N \trianglelefteq G$, and if $P$ is a $p$-Sylow subgroup of $N$ with $P\trianglelefteq N$, then $P\trianglelefteq G$. I understand the proof given in my book ...
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60 views

Non-abelian solvable group of order $p^2q$ that including no subgroup of order $pq$.

I am working on a non-abelian solvable group of order $p^2q$ that including no subgroup of order $pq $. My problem is this: Does this group have a normal sylow $p$-subgroup? In the case that $p>...
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25 views

Topological generator of profinite group

I'm trying to prove that $\pi(\operatorname{Spec}\mathbb{Z}[\frac{1}{2}])$ is topologically generated by its $2$-Sylows (Exercise 6.30 of Lenstra's notes on Galois Theory for Schemes). I've been given ...
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1answer
22 views

Question on uniqueness of minimal normal subgroup

Let $H$ be minimal normal $p$-sylow subgroup of a group $G$. Order of $G$ is $ap^{m}$ where gcd$(a,p^m) = 1$ Since $H$ is normal $p-SSG$ it implies that is unique of that order i.e. $H$ is the only ...
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2answers
79 views

Proving a group of order $35^3$ is solvable

Is my thinking correct when asked to show that a group $G$ of order $35^3$ is solvable, I first show that by the sylow theorems there exists a sylow $p$-subgroup of order $5^3$ and another unique ...
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2answers
66 views

If abelian $P\in{\rm Syl}_p(G)$, and $H\le P$ and $H^g\le P$, show $g\in G$ is the left product of an element of $N_G(P)$ with an element of $C_G(H)$

If abelian $P \in{\rm Syl}_p(G)$, and $H \leq P$ and $H^g \leq P$, show $g \in G$ is the left product of an element of $N_G(P)$ with an element of $C_G(H)$ First, I set $g=nc$, where I hope to find ...
3
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1answer
61 views

Prove that if a finite group $G$ is soluble and has more than one $17$-Sylow subgroup, then it has more than $100$ $17$-Sylow subgroups.

Prove that if a finite group $G$ is soluble and has more than one $17$-Sylow subgroup, then it has more than $100$ $17$-Sylow subgroups. My main reason for asking: Do people find this easy or hard? ...
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1answer
68 views

Difficulty using Sylow theorems

I am really struggling to apply Sylow's theorems to various problems, so any help is much appreciated. For part (a) here, since we are supposing that $H<G$, I tried to use an element $g$ of $G$ ...
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1answer
58 views

Trouble using Sylow's theorems

So, I found part (a) fairly easy, and also managed part (c) assuming (b) to be true. Any help with (d) and especially (b) would really be appreciated. My issue with part (b) is understanding which of ...
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1answer
34 views

Every finite group $G$ such that $p\mid |G|$ has a maximal p-subgroup

I'm reading the book "An Introduction to the Theory of Groups" by Joseph Rotman. Maximal $p$-subgroup is defined as: $P$ is a maximal $p$-subgroup of a group $G$ if for every $Q\le G$, $Q$ ...
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2answers
49 views

How many equivalence classes are there in $J$.? [closed]

Let $G$ be the symmetric group $S_5$ of permutations of five symbols. Consider the set $J$ of subgroups of $G$ that are isomorphic to the non-cyclic group of order $4$. Let us call two subgroups $H$ ...
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29 views

Are p-Sylow subgroups of a profinite group dense?

I'm trying to show that $\pi(\operatorname{Spec}\mathbb{Z}\left[\frac{1}{2}\right])$ is topologically generated by its 2-Sylow. I already know that the fundamental group of $\operatorname{Spec}\mathbb{...
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2answers
72 views

Let $G$ a nilpotent group such that $10$ divides $|G|$. It is true that $G$ has element of order $10$?

Let $G$ a nilpotent group such that $10$ divides $|G|$. It is true that $G$ has element of order $10$? We know that $G$ can be expressed like direct sum of Sylow's subgroups and that $G$ has a normal ...
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1answer
51 views

Classification of groups $G$ of order 24.

I supposed $n_3=4$ and $n_2=3$, I want to show that $G\cong S_4$. Then I made $G$ act by conjugation on $\text{Syl}_4 (G)=\{X_1,X_2,X_3,X_4\}$. This action defines a representation $\varphi:G\...
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34 views

Show that every group of order $4125=3\cdot 5^3\cdot 11$ is solvable.

Show that every group of order $4125=3\cdot 5^3\cdot 11$ is solvable. Proof: Suppose $G$ is a group of order $4125$. By Sylow's Theorems, $n_3 \equiv 1 \mod 3$ and $n_3 | 5^3\cdot 11$ $\...
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1answer
47 views

Show that any group of order 294 is solvable.

Show that any group of order 294 is solvable. Proof: Suppose $G$ is a group of order $294=2\cdot 3\cdot 7^2$. By Sylow's Theorem, $n_2\equiv 1 \mod 2$ and $n_2|147$ $\Rightarrow$ $n_2=1,3,7,21,49,147$...
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1answer
57 views

Doubts on proving there are only $2$ abelian groups of order $12$, up to isomorphism?

I have been tasked with the following exercise: Prove there are only $2$ abelian groups of order $12$, up to isomorphism. I have read some texts and watched some videos but got very confused. I want ...
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1answer
64 views

Given a finite solvable group $G$, prove that a minimal normal subgroup $H$ is a $p$-group

Given a finite solvable group $G$, and a minimal normal subgroup $H$, prove that $H$ is a $p$-subgroup for some prime $p$. My Attempt: I am trying to write this proof without using the term "...
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1answer
45 views

What's the geometrical interpretation of the transitive, faithful action of $S_4$ by conjugation on the set of its four Sylow $3$-subgroups?

A pair of recent posts (here and here) have posed the same question: show that the action of the symmetry group of a cube on pairs of opposite faces defines a surjective homomorphism from $S_4$ to $...
2
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1answer
48 views

Does a group of order 24 have a sylow 11 subgroup

I am going through Algebra In Action by Shahriari and on page 147 it says Definition 7.8 (Sylow subgroups). Let G be a finite group with |G| = $p^a$m, where $p \nmid m$, and $a$ is a non-negative ...
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1answer
28 views

Notation for a proof containing group actions

I am writing a proof to Sylow's 2nd Theorem. I have done it using group actions. My tutor told me it would be wise to introduce notation to make it more clear what group action I am talking about. I ...
2
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1answer
73 views

Proof of Sylow's Theorem (Herstein) - why is $no(H) = o(G)$?

The theorem is: (Sylow's theorem): If $p$ is a prime number, and $p^\alpha |o(G)$, then $G$ has a subgroup of order $p^\alpha$. Right before the proof, the author has established that if $n = p^\...
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0answers
51 views

Prove that $G$ is solvable group without using Burnside Theorem.

Let $G$ group of order $5^2q^m,\ m>1$, with $5\neq q$ and $q$ prime. We want to show that $G$ is solvable without using Burnside theorem. If $P\in \mathrm{Syl}_{q}(G)$ and $n_{q}=[G:N_{G}(P)]$ we ...
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1answer
26 views

Sylow 2-subgroup and Sylow 3-subgroup of $D_{24}$ (of order $48$).

Find Sylow 2-subgroup and Sylow 3-subgroup of $D_{24}$ I have found $n_2 = 1$ or $3$ and $n_3 = 1$ or $4$ or $16$, where $n_2$ & $n_3$ are the number of Sylow $2$-subgroups and Sylow $3$-...
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3answers
48 views

$p$-subgroup of Sylow and Normalizer

Let $G$ be a finite group, $p$ a prime that divides $|G|$, and $P$ a $p$-subgroup of Sylow of $G$. Show that $P$ is the unique $p$-subgroup of Sylow that is in $N_G(P)$. I tried this by assuming there ...
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1answer
35 views

Sylow subgroup of a group generated by two subgroups

Let $G$ be a finite group and $H=\langle A ,B\rangle$ be a subgroup of $G$ generated by subgroups $A$ and $B$ of $G$. Is it true that $H_r=\langle A_r,B_r\rangle$, where $H_r$, $A_r$ and $B_r$ are ...
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1answer
13 views

Sylow theorem in fraighle's text

I've been struggling with the first Sylow theorem. Then γ−1[K] is a subgroup of N[H] and hence of G. This subgroup contains H and is of order pi+1 How come it has order pi+1, Couldn't understand that ...
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2answers
50 views

$\langle A,B \rangle$ is the unique Sylow $2$-subgoup of $SL_2(F_3)$

Prove that the subgroup of $SL_2(F_3)$ generated by $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ and $\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$ is the unique Sylow $2$-subgoup ...
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121 views

Intuition behind picking group actions and Sylow

A common strategy in group theory for proving results/solving problems is to find a clever group action. You take the group you are interested in (or perhaps a subgroup), and find some special set ...
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1answer
31 views

For every $p$-subgroup $Q$ there exists a $p$-Sylow $P$, such that $Q \leq P$

I am trying to prove a consequence of Sylow's theorems. Let $G$ be a group with $|G|=p^km$, where $p$ is a prime and $k, m$ are integers such that $p$ does not divide $m$. Let $Q$ be a $p$-subgroup of ...
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1answer
22 views

Properties of normalizer of (Sylow $p$-subgroup of the stablizer of a point)

(The original is a proof exercise but apparently contain print errors so I had to edit it to make sense, thus I'm not sure about its validity, which greatly hindered my attempt) Exercise 2.4.13: Let $...
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1answer
86 views

A group of order $150$ has at least $4$ conjugacy classes made by elements of order a power of $5$?

Let be $G$ a group of order $150$ and $g \in G\,\,$ s.t. $|g|=25\,$: Prove that $\exists \,\,K\lhd G\,\,$s.t. $K \simeq C_5$. $\,\,$ Say whether $\exists \,\,h \in G\,\,$ s.t. $|h| =15$.$\,\,$ Prove ...
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1answer
45 views

A problem about Sylow Theorem [closed]

I'm learning Sylow Theorem. My textbook uses the following result directly and does not tell me why. I don't know how to prove it. $C_{p^rn}^{p^r}\equiv n \mod pn$. Here $p$ is a prime number and $r,n$...
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1answer
49 views

$|G| = pqr^2$, $G$ not simple [closed]

If $G$ is a group with cardinality $pqr^2$ and $p,q,r$ different primes $\ge 2$, and given $1+np$ doesn't divide $qr^2$ for any $n \in \mathbb{Z}^+$, How can you show $G$ is not simple? I feel like ...
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1answer
37 views

$2$-Sylow of a subgroup of $A_n$ [closed]

I don't know how to start to prove this: Let $H$ be a subgroup of $S_n$ and let $T$ be a subgroup of $H$ such that $T$ is a $2$-Sylow of $H$. Show that $H$ is conntained in $A_n\iff$ $T\le A_n$. Maybe ...
1
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1answer
58 views

Decomposition of a permutation group into a direct product

Let $S_6$ be the symmetric group of a set of six elements. Let $\sigma=(1,3,4,6)$ and $\tau=(1,3,4,6)(2,5)$ be two cycles in $S_6$. (A) Determine the order of the two permutations and the order of $G:=...
3
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0answers
24 views

Existence of subgroups in group of order $2n$ [duplicate]

Let $G$ be a finite group with order $2n$. Let's assume $G$ has $2$ distinct subgroups $G_1$ and $G_2$ both of order $n$. I want to prove that there exists another subgroup $G_3$ of order $n$ with $...
0
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0answers
12 views

Which group is simple? [duplicate]

I am currently looking whether a group G of the following order is simple or not $ord(G)=100$ $ord(G')=200$ We have first that for $G$, $ord(G)=2^25^2$. For $G'$ we have that $ord(G')=2^35^2$. Now ...
3
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2answers
51 views

Application of Sylow to a group of order 60

Let $G$ be a group with $|G|=60=2^2 \cdot 3 \cdot 5$ and assume $G$ has a unique $3$-Sylow subgroup $H_3$. The claim is that then $G$ has also a unique $5$-Sylow subgroup. This is the procedure to ...
1
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1answer
57 views

Conjugation action on a semi-direct product $N\rtimes H$

Let $G$ be a finite group, with $H\le G$ and $N\trianglelefteq G$ such that $G=N\rtimes H$. Suppose that the conjugation action of $H$ on $N$ induces $2$ orbits in $N$. Prove the following requests: ...
2
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2answers
60 views

A question about Sylow's theorems and index of groups.

I'm trying to prove this exercise: Let be $G$ a finite group, p a prime number, and $H$ a normal subgroup. If $p \nmid [G:H]$, then $\{ x \in G : o(x)=p^{n_x}, \text{ with } n_x \in \mathbb{N} \} \...
3
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1answer
56 views

Group of order $pqrt$

Let $G$ be a group of order $pqrt$, where $p>qrt$ and $p,q,r,t$ are prime ($\underline{\text{not necessarily distinct}}$). I have to show that $\exists$ $M,N,L \le G$ of order $pq,pr$ and $pt$. ...
2
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2answers
57 views

inner semidirect and outer semidirect relationship

https://kconrad.math.uconn.edu/blurbs/grouptheory/group12.pdf. In the text above, the author explains how to find all groups of order 12. He does so by showing that a group $G$ (of order $12$) is ...

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