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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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1answer
31 views

$|G|=p^2q^2$ where p,q are primes, $p<q$ and $p \nshortmid (q^2-1)$ Show $G$ is Abelian.

$|G|=p^2q^2$ where p,q are primes, $p<q$ and $p \nshortmid (q^2-1)$ Show $G$ is Abelian. ($p$ does not divide $q^2-1$) The first thing to note is that $(q^2-1)=(q-1)(q+1)$ so we can conclude that ...
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1answer
36 views

How to prove $Q_8$ is not subgroup of $S_4$ by sylow thm?

Prove $Q_8$ is not subgroup of $S_4$ Hi. It is trivial that $D_4$ is a subgroup of $S_4$. But the case $Q_8$ make me confuse why this group is not a subgroup of the $S_4$. Why is the $Q_8$ isn't ...
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0answers
29 views

I have two confusion in sylow theorem formula . True/false

I have two confusion in sylow theorem formula $1.$Order of a $n$ sylow subgroup in $GL_n (\mathbb{F_p})= \frac{\prod_ {i=0}^{n-1} (p^n -p^i)}{(p-1)^n p^{\frac {n(n-1)}{2}}}$ $2.$ Order of a $q-$...
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1answer
33 views

There are no simple groups of order 240

I'm trying to prove that there are no simple groups of order $240$. So let $G$ be a simple group such that $|G|=240=2^4\cdot3\cdot5$. Then $$n_2\in\{1,3,5,15\}\quad n_3\in\{1,4,10,40\}\quad n_5\in\{1,...
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1answer
41 views

Finding a $p$-Sylow-subgroup of GL$_2(\mathbb{Z}/p\mathbb{Z})$ [closed]

Can somebody help me to find a $p$-Sylow-subgroup of GL$_2(\mathbb{Z}/p\mathbb{Z})$? I actually dont even know how to start :/ Thank you!
3
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0answers
41 views

What does the abelianization of a Sylow $2$-subgroup of the symmetric group look like?

I started thinking about this for no particular reason. Let $P_n$ be a Sylow $2$-subgroup of the symmetric group $S_{2^n}$. What does its abelianization $P_n/[P_n,P_n]$ look like? The groups $P_n$ ...
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0answers
46 views

What does Sylow theory have to say about group presentations?

What does Sylow theory have to say about group presentations? Of the books on combinatorial-group-theory I have looked in so far, the following do not contain any reference to Sylow's Theorems: ...
7
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2answers
137 views

Construct a nonabelian group of order 44

Let $G$ be a group s.t. $|G|=44=2^211$. Using Sylow's Theorems, I have deduced that there is a unique Sylow $11$-subgroup of $G$; we shall call it $R$. Let $P$ be a Sylow $2$-subgroup of $G$. Then we ...
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2answers
47 views

Do there exist (non-trivial) prime solutions to the equations $p^2 = 1$ mod $q$, $q = 1$ mod $p$?

Question: Do there exist odd primes $p$ and $q$ such that $$p^2 = 1 + qt,\quad q = 1 + ps$$ for some positive integers $s,t$? I've written some code which has verified that no solutions exist for $p,q ...
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1answer
546 views

Misunderstanding of Sylow theory

I think I have a misunderstanding a part of Sylow theory for groups or I have made a big mistake in my reasoning below. We have the following lemma in Sylow theory: Let $G$ be a finite group and ...
0
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1answer
62 views

Prove $G=N_G(P)H$ where $H \lhd G$ and $P \in Syl_p(H)$. [duplicate]

Prove $G=N_G(P)H$ where $H \lhd G$ and $P \in Syl_p(H)$. I definitely need some help with this one. I was thinking that maybe the problem meant to say that $P$ was a sylow subgroup of $G$, that would ...
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0answers
23 views

Group of order $105$ with a normal Sylow $3$-subgroup is cyclic [duplicate]

Let $G$ be a group of order $105 = 3 \times 5 \times 7$ with a normal Sylow $3$-subgroup, prove that $G$ is cyclic. My attempt I've been able to show that $G = NP$, where $N$ is the normal Sylow $3$-...
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0answers
44 views

Proving every group of order 2673 has a non-trivial proper normal subgroup

If $|G|=2673=3^511$ then let $S \in Syl_3(G)$. Then the number of conjugates of $S$, $n_s \equiv 1$ mod $3$ and $n_s | 11$. But the only divisors of 11 are 1 and 11, and therefore $n_s=1$ so the $S$ ...
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1answer
32 views

Why is the number of Sylow 2 subgroups of simple group with order 60 not able to be 1 or 3?

I want to show that a simple group of order 60 is isomorphic to $A_5$. In the process, I am stuck at the part in which I have to show that the number of Sylow 2 subgroups (whose orders are 4) cannot ...
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0answers
58 views

Sylow Subgroups of the Rubik's Cube Group

The Rubik's Cube Group has order $2^{27} 3^{14} 5^3 7^2 11$. What is known about the Sylow subgroups of this group? Do they have an intuitive meaning with regards to the symmetry of the cube?
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37 views

Geometric reason why there is a unique Covering space corresponding to Sylow Subgroup

By the Galois Correspondence (for path connected, locally simply connected spaces $X$, say) if $\pi_1 (X)$ is a finite group, then there is a unique isomorphism class of non-based covering space with ...
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1answer
32 views

Small step in a Sylow theorem proof

I'm following https://kconrad.math.uconn.edu/blurbs/grouptheory/sylowmore.pdf and in particular the following paragraph: Let $N = N_G(P)$ be the normalizer of $P$ in $G$. Then all the elements of ...
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2answers
38 views

Using the number of Sylow 2-subgoups of $G$.

I want to show that a group of $G$ of order $56$ is not simple using the number of Sylow $2$-subgroups of $G$ with $n_{2}=7$ and considering two Sylow $2$-subgroup $P_{1},P_{2}$ of $G$. (I know the ...
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1answer
98 views

Problem 2B.4 in Finite Group Theory by Issacs

Suppose a finite group $G$ has more than one Sylow-2 subgroup and any two intersects trivially. Show $G$ contains exactly one conjugacy class of involutions. Here are some of my thoughts: Suppose not,...
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1answer
76 views

How to show that $G$ can be expressed as a semidirect product

Let $G$ be a group of order $42$. Prove that $G$ is a semidirect product of a normal subgroup of order $21$ and $\mathbb{Z}_2$. My attempt: $G$ has unique Sylow 7 subgroup and Sylow 3 subgroup is not ...
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3answers
38 views

A question about the consequence of Sylow's theorems

The three Sylow theorems are stated here. I don't understand this statement "A consequence of theorem 3 is if $n_p = 1$, then the Sylow p-subgroup is a normal subgroup". I understand that if $n_p = ...
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0answers
74 views

Group of square free order with a normal $p$-Sylow is solvable

Let $G$ be a group of order $p_1...p_s$ where $p_1,...,p_s$ are distinct primes. If $G$ has a normal $p$-Sylow subgroup, then $G$ is solvable. We proceed by induction on $s$. If $s = 1$, $G$ is ...
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0answers
66 views

Group of order $108$ not simple using Sylow theorems on Sylow-$2$-subgroups

Show that any group of order $108$ is not simple. I can show this using Sylow theorems on Sylow-$3$ subgroups. I was not able to completely justify for Sylow-$2$ subgroups though. In case of Sylow-$...
4
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1answer
73 views

Group of order $5^k\cdot 8$ has normal subgroups of order $5^{k},5^{k}\cdot2,5^{k}\cdot4$

Let $G$ be a group of order $5^k\cdot 8$. I was trying to prove that there are normal subgroups of order $5^{k},5^{k}\cdot2,5^{k}\cdot4$. I saw the following statement: Let $P$ be a $p$-Sylow ...
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0answers
52 views

Every group of order 399 is abelian and not simple

Let $G$ a group with order $399$ = ${3*7*19}$, from Sylow-theorem, I know $n_{3}$ must be $1$ or $7$ or $19$ or $133$, for $n_{7}$ must be $1$ or $57$ and for $n_{19}$ must be 1. I know $n_{19}$ is $1$...
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1answer
33 views

$p$-group problem

Let $A,B,C$ are three subgroups in a way that $1<A \triangleleft B \triangleleft C$. With $B/A$ and $C/B$ are $p$-groups. Then prove that $|C|$ is also a $p$-group. I have been trying to prove it ...
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1answer
38 views

How to prove this property of this group of order $20$ without the Sylow theorems?

In Artin's Algebra under the section on the Class Equation is the exercise The class equation of a group $G$ is $1+4+5+5+5$. (a) Does $G$ have a subgroup of order $5$? If so, is it normal? (b) Does ...
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2answers
40 views

$n$ Sylow $p$-subgroups of $G$ $\implies$ $\exists H<Sym_n$ s.t. $H$ has $n$ Sylow $p$-subgroups?

Let $G$ be a finite group having exactly $n$ Sylow $p$-subgroups for some prime $p$. Show that there exists a subgroup $H$ of the symmetric group $S_n$ such that $H$ also has exactly $n$ Sylow $p$-...
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3answers
78 views

How do I know that $\mathbb{Z}_{175}$ is not an additional subgroup of order $175$?

Here was the original problem statement. Enumerate all non-isomorphic groups of order $175$. See that $|G| = 175 = 5^2\cdot7$. Therefore by Sylow's first $H \leq G$ & $|H| = 25$ in addition to ...
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1answer
139 views

How to show that, in this case, all the Sylow $p$-subgroups of $G$ are abelian.

Let $G$ be a finite, simple group of order $n$. Let $p$ be a prime divisor of $|G|$ and suppose that the number of conjugacy classes of $G$ is $> \frac{n}{p^2}$. Then all the Sylow $p$-subgroups of ...
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1answer
48 views

Is the intersection of Frattini subgroup and a Sylow subgroup contained in the Frattini subgroup of the Sylow subgroup?

Suppose $G$ is a finite group, $P$ is a Sylow p-subgroup of $G$. Is it always true, that $\Phi(G) \cap P$ is a subgroup of $\Phi(P)$? Here $\Phi(G)$ is the Frattini subgroup of $G$. I managed to ...
5
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1answer
289 views

Sylow 2 subgroups of S4

I am trying to find all the Sylow 2 subgroups of S4 using Sylow’s theorems. Now, I know that a Sylow 2 subgroup of S4 has size 8, and that there are either 1 or 3 of them (as the number of of Sylow 2-...
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1answer
43 views

Finding number of subgroups of order 3 in an abelian group of order 24

Problem Find the number of subgroup of order 3 in an abelian group of order 24? Attempt Let $n_3$ be the number of subgroup of order 3 . Then by Sylow's theorem $n_3|8$ and $n_3 \equiv 1mod 3$. From ...
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1answer
47 views

Group with order $30$ is not a simple group.

Could someone prove the sentence given above in the title? I know that Sylow theorems should be used here. Let $N_{p}$ stands for number of $p$-Sylow subgroups in group $G$. I tried to use ...
3
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1answer
46 views

Group of order $p^{\alpha}q$ is not simple.

$|G|=p^{\alpha}q$, where $p,q$ are distinct primes, $\alpha \geq 1$. Show $G$ is not simple. I am trying to follow a proof and I understand all of it except one part which is blocking me. The proof ...
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1answer
85 views

Group of order 60

Let $G$ be a simple group of order $60$: i) Find the number of Sylow $3$- and $5$-subgroups of $G$. ii) Show that the alternating group $A_5$ has a subgroup of order $12$. iii)Show that $G$ is ...
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1answer
75 views

Why does $a^p \equiv 1\ (\text {mod}\ q)$?

Suppose $G$ is a group of order $pq$ with $p<q, p \nmid q-1$ and $p,q$ are primes. Then $G$ is cyclic. The way our instructor has proved this theorem is as follows $:$ He first proved that $G$ ...
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2answers
84 views

Sylow Subgroups and Conjugation

Suppose that $G$ is a finite group whose order is divisible by a prime $p$. Let $S$ be the set of Sylow $p$-subgroups of $G$; let $H$ be an element of $S$. $H$ acts on $S$ by conjugation. The fact ...
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0answers
87 views

Find the number of $2$-Sylow subgroups in $GL_2(\mathbb F_5)$

First, we can calculate the order of $GL_2(\mathbb F_5)$ to find the order of the $2$-Sylow subgroups. $$|GL_2(\mathbb F_5)|=(5^2-1)(5^2-5)=24\cdot 20=2^5\cdot 3\cdot 5$$ Thus we have $|P|=2^5$. ...
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1answer
43 views

The normalizer of a $p$ sylow subgroup is itself

By the orbit-stabilizer theorem since the action is transitive then an orbit $\{gPg^{-1}: g\in G\} =n_p$ is equal to the number of $p$ sylow subgroups in a group $|G|=p^{\alpha}s$ with $(p^\alpha,s)=1$...
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2answers
109 views

Show that a group of order $90$ is solvable

Let $G$ be a group of order $90$. Show that $G$ is solvable. Given that $90=2 \cdot 5 \cdot 3^2$ is of the type $pqr^2$, with $p, q,$ and $r$ primes, this case is not straightforward. My attempt to ...
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1answer
24 views

In a group of order 400, is the normalizer of one of the 16 Sylow 5-subgroups itself?

In a group $G$ of order $400 = 2^4 \cdot 5^2$, assume there are sixteen Sylow 5-subgroups. Let $P_5$ be one of them. Is the normalizer $N_G(P_5)=P_5$? I think this is true, as the order of $P_5$ is ...
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2answers
106 views

Show that a group of order 66 has a normal subgroup of order 33.

This question is somewhat similar to: A group of order $66$ has an element of order $33$. However, I do not understand how I would show that the subgroup of order 33 is normal. So far I have that ...
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0answers
49 views

If $P$ is a Sylow $p$-subgroup of $G$ then there exists a normal subgroup $K$ such that $G/K$ is isomorphic to $P/P \cap G'$

In 'Finite Groups' by Gorenstein, it is stated that if $P$ is a Sylow $p$-subgroup of $G$ then there exists a normal subgroup $K$ such that $G/K$ is isomorphic to $P/P \cap G'$′. The proof is the ...
4
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1answer
53 views

Abelian normal subgroups of A-groups

Let $G$ be a finite solvable group, where every Sylow subgroup is abelian. I want to show that if $A\lhd G$ is an abelian normal subgroup, then $$ A=(A\cap Z(G))(A\cap G')$$ This is easy if $A$ is a ...
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0answers
43 views

Show that any finite nilpotent group of square free order is cyclic.

Show that any finite nilpotent group of square free order is cyclic. Hint: Suppose G is such a group. Any Sylow subgroup of G is of prime order. Hint: Any finite nilpotent group is the direct ...
2
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1answer
80 views

How can $SN_G(D)=G$ if $S$ is not normal in $G$?

Is there any theorem which says anything like $SN_G(D)=G$ where $S$ is a $p$-Sylow subgroup and $D$ is the intersection it has with some other subgroup? I know the first that will probably spring to ...
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0answers
91 views

trying to understand this proof of sylows theorem that say the number of p-sylow subgroups is 1+kp

I'm Very confused as to what my lecturer means in the final few lines of a proof of one of sylows theorems means. The theorem in question is the one that says the number of sylow P-subgroups is 1+kp. ...
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0answers
58 views

a few questions about what's going on in this proof of sylow's theorem I found

Note: If someone wants to even just answer my first question in the comments until someone else decides to give a full answer I'd be pretty happy. I just want to know there's no mistakes in it before ...
0
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1answer
26 views

In a group of order $m p^n$ for $p$ prime, if $k<n$, is there an element of order $p^k$? [duplicate]

Let $G$ a group of order $mp^n$ where $p$ is prime. Let $k\leq n$. Is there an element of order $p^k$ ? Since $p$ divide $|G|$, by Cauchy theorem, there is $g\in G$ s.t. $g$ has order $p$. I can't do ...