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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-2
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0answers
48 views

There are 267 not isomorphic groups of order 64. How many of these are Abelian? Describe them. [on hold]

There are 267 not isomorphic groups of order 64. How many of these are Abelian? Describe them.
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2answers
22 views

Group order 168 having a normal subgroup of order 4, then G has a normal subgroup of order 28.

Prove 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. resolution: P4 is the ordeme subgroup 4 and P7 the order subgroup 7. As P4 is ...
2
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2answers
17 views

Need help in understanding a certain step of a certain proof in finite group theory and group actions

A proof is from Aluffi's textbook "Algebra: Chapter 0". A statement: There are no simple groups of order $24$. The proof from the book: Let $G$ be a group or order $24 = 2^33$, and consider ...
2
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1answer
37 views

On the maximum number of Sylow subgroups

I have some doubts regarding the first sentences below the fourth case in the more elementary solution by Prof. Samuel to problem 11856 of the Monthly. Here you have a screenshot of the relevant part ...
1
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2answers
36 views

Sylow subgroup related

In a group of all invertible $4 \times 4$ matrices with enteries in the field of $3$ elements, What is the cardinality of any $3$- sylow subgroup?
1
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2answers
35 views

If $a$ has order $p$ and $aPa^{-1}=P$ then $a\in P$

If $G$ is a finite group, $P$ a $p$-Sylow subgroup of $G$, $a\in G$ has order $p$ and $aPa^{-1}=P$ then $a\in P$. This is proved in Rotman: Proof: We have $a\in N_G(P)$. If $a\notin P$ then $aP\in ...
3
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3answers
77 views

Is there a non trivial normal subgroup of a group $G$, where $|G|=pm, \ \gcd(p,m)=1$?

In fact, the exercise I'm in is this: Suppose you have an irreducible polynomial $f(x)\in \mathbb{Q}[x]$ of degree $p$, where $p$ is a prime. Also suppose that $K$ is a splitting field of $f(x)$ over ...
1
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1answer
25 views

Counting the number of distinct elements in Sylow subgroups if $|G|=30$

I'm trying to prove that if $|G|=30=2\cdot 3\cdot 5$ then $G$ has a normal $3$-Sylow or a normal $5$-Sylow. By the Sylow theorems, we would argue by contradiction in order to prove it cannot be $n_3=...
1
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2answers
89 views

Let $G$ be a group and $N$ be a minimal normal subgroup of $G$, $G_p \in Syl_p(G)$, where $p$ is an odd prime divisor of $|G|$.

Let $G$ be a group and $N$ be a minimal normal subgroup of $G$, $G_p \in Syl_p(G)$, where $p$ is an odd prime divisor of $|G|$. Suppose that $M/N$ be a maximal subgroup of $G_pN/N$. then $M = PN$ for ...
1
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3answers
74 views

Sylow $p$-subgroup of a finite group

I know: Let $P$ be a Sylow $p$-subgroup of a finite group $G$.If $N$ is normal in $G$, then $P \cap N$ is a Sylow $p$-subgrup of $N$. But if $N$ is not normal in $G$ , there is also the issue? ...
3
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1answer
70 views

Questions about $\mathrm{SL}_2(\mathbb{F}_7)$

Let $G=\mathrm{SL}_2(\mathbb{F}_7)$, which has order $336=2^4\cdot 3\cdot 7$. And I may assume that $G$ is generated by the two matrices $$\begin{pmatrix}1&1\\0&1\end{pmatrix}, \begin{pmatrix}...
0
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1answer
38 views

Image of Sylow $p$-subgroup is Sylow $p$-subgroup

If $f:G\to H$ is an epimorphism between finite groups and $K\subset G$ a Sylow p-group then I want to show that $f(K)\subset H$ is also a Sylow p-group. So we want to show that $|f(K)|=p^m$ where $m$...
0
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2answers
27 views

If $N_G(P)\le H$ then $H=N_G(H)$

If $G$ is a group, $P$ is a Sylow subgroup of $G$ and $N_G(P)\le H\le G$ then $H=N_G(H)$. I solved this in case $G$ is finite: $P$ is also a Sylow subgroup of $H$ and $H$ is a normal subgroup of $...
0
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1answer
33 views

If $P_1,P_2,P_3$ are $p_i$-Sylows then $P_1\cap P_2P_3=\{e\}$

If $P_1,P_2,P_3$ are normal $p_i$-Sylows of a group $G$ ($i=1,2,3$ and $p_i$ distinct primes) then $P_1\cap P_2P_3=\{e\}$. Every element of $P_1$ has order some $p_1^n$. So I wanted to show every ...
2
votes
1answer
39 views

Showing a surjective homomorphism maps Sylow $p$-groups to Sylow $p$-groups [duplicate]

If $f:G\to H$ is a surjective homomorphism of finite groups, then $f$ sends Sylow $p$-subgroups to Sylow $p$-subgroups. Here's what I have. Suppose $\vert G \vert=p^km$ with $(p,m)=1$. Let $P\in \...
1
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1answer
23 views

$N_G (P \cap Q) $ has more than one Sylow p-Subgroup under some conditions.

Let $p$ be a prime number. Let $G$ be a group with more than one Sylow p-subgroup Over all pairs of distinct Sylow p-subgroups Let P and Q be chosen so that |P$\cap$Q| is maximal. I want to prove ...
1
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2answers
70 views

All groups of order 12; Dic12 and D6

I know this has been asked in various forms before, but so far I have failed to understand those answers properly. I've also read several papers discussing this, but I don't really get it. I have an ...
0
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0answers
28 views

Sylow theorems and normality

Let |G|=14924=2^2.7.13.41 We can see Sylow 41-subgroup is normal But when I try for Sylow 7 and 13, I face some problems. I use the argument following: Let P be a Sylow 41 and R be a Sylow 7 subgroup....
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0answers
24 views

Let $G$ be a $p$-group, with $|G|= p^n$. Show that $G$ has a normal subgroup of order $p^m$ for each integer $0 < m < n$. [duplicate]

I think I have solved a problem using one of the sylow theorems. But, if this proof is correct, I think I've cheated a little. Since the chapter on Sylow theorems comes directly after the chapter on ...
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0answers
29 views

Why is a nontrivial finite group is nilpotent if every maximal subgroup is normal?

In the following proof I understand the above proof up to the part where it says "by Sylow theory $N(M)=M$", could someone explain to me why is this true. We have just started learning about group ...
0
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0answers
13 views

Injective map from the Cartesian product of two sylow-p-subgroups into the group.

Let $G$ be a group of order 148. Show that $G$ is not simple. The given solution goes as follows: $148 = 4 × 37$. By Sylow’s theorem, it has at least one subgroup $P$ of order 37. If $P'$ is another,...
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1answer
18 views

The order of the normalizer of a $p$-subgroup of $S_{p}$ [closed]

I found it In Exercise in abstract algebra by Dummit and Foote. Let $P$ be a Sylow $p$-group of $S_p$. What is the order of $N_{S_p}(P)$?
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1answer
35 views

Groups of order 12 with a normal 3-subgroup contain an element of order 6

Let $G$ be a group of order $12$ with a normal $3$-subgroup (which is unique by Sylow's theorems). Does it contain an element of order $6$? I just need a hint to prove it without classifying all the ...
1
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2answers
33 views

Are the normalizers of Sylow p-subgroups isomorphic?

Let $G$ be a finite group and $A,B \in \text{Syl}_p G$, for some prime $p$. Is it always true that the normalizers $N_G(A)\cong N_G(B)$? I just need a hint to get started, because I don't know where ...
1
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1answer
90 views

Group of order $p^2+p $ is not simple [duplicate]

Can some one please give me a hint to prove that every group of order $p^2+p$ is not simple?
0
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1answer
31 views

Cardinality of a power set? Or is “all subsets of a set” $\neq$ power set?

Again as usual, group theory is muddling me up. A proof of the Sylow $I$ theorem starts as follows Let $X$ be the set of all subsets of $G$ with $|A|=p^m$. where, the setting I have for Sylow $I$...
0
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2answers
35 views

Non-abelian, simple subgroups of $S_n$

I am trying to prove, as part of a larger theorem, that if $G$ is a non-abelian finite, simple group of order $>2$ and $G$ is a subgroup of $S_n$, then $G$ must be a subgroup of $A_n$. Any ideas ...
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votes
1answer
80 views

Classify groups of order 100 [closed]

So I am currently trying to Classify all groups of order 100 through an extensive proof; and this is as far as I have gotten so far, wondering how to go beyond the fact that both squares (Z4 & Z25)...
1
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0answers
64 views

When can a dihedral group $D_{n}$ of order $2 n$ be a $p$-group?

A $p$-group is a group where the order of every group element is a power of the prime $p$. The presentation of a dihedral group $D_n$ of order $2 n$ is as follows. $$D_n = \langle x, y \mid x^n = y^2 ...
5
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1answer
51 views

Group with exactly six Sylow 5-subgroups?

Give an example of a group with exactly six Sylow 5-subgroups. I think $A_5$ works because it has 6 subgroups of order 5: $\langle(12345)\rangle,\langle(12354)\rangle, \langle(12435)\rangle, \...
2
votes
1answer
47 views

Square of order of a Sylow p-subgroup in the nonabelian simple groups

Is it true that for all Sylow subgroups $P$ of a nonabelian simple group $G$ that $|P|^2 < |G|$? If $P$ is abelian, this is an easy consequence of Brodkey's theorem (Suppose that a Sylow $p$-...
0
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0answers
58 views

Number of $2$-Sylow subgroups of $S_5$

Find the number of $2$-Sylow subgroups of $S_5$ and represent one of them. Would someone please give a hint for how to start?! I only can say that it should be an odd number dividing $5!$ (these can ...
0
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0answers
45 views

I need help proving that a certain subgroup is a direct product of specific subgroups

Let G be a group, P be an abelian Sylow-p subgroup of G. Let $N=N_G(P)$ and assume that H is a complement of P in N which I believe means that $HP=N$ and $H\cap{P}=1$ Prove that $P = P_1 \times P_2$ ...
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1answer
33 views

If The order of $G$ is $pm$ and $p$ is prime such that If $H$ is normal to $G$ with order $p$ show that H is characteristic

I will be grateful for your help If The order of $G$ is $pm$ and $p$ is prime such that If $H$ is normal to $G$ with order $p$ show that H is characteristic
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0answers
12 views

A $p$-subgroup of a finite group is either a Sylow $p$-subgroup or properly containted in a Sylow $p$-subgroup of its normalizer

This exercice is from Aschbacher's book "Finite group theory". I am not asking for a complete solution, just for a hint. Here is a partial solution, when the ambient group $G$ is a $p$-group. If $X$ ...
0
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2answers
41 views

Does $S_6$ have an abelian sylow $2$ subgroup.

How do I check if $S_6$ has an abelian sylow 2 subgroup. Order of any sylow 2 subgroup is $16$ and by sylows theorem it has $45$ sylow 2-subgroups, but how to check whether it has any abelian sylow 2-...
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0answers
38 views

Show that the number of conjugates of a p-Sylow subgroup is not divisible by p

I'm trying to solve the following problem: "Let $P$ be a $p$–Sylow subgroup of a finite group $G$. Consider the set of conjugate subgroups $gPg^{-1}$ with $g \in G$. Show that the number of ...
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1answer
30 views

If $S$ is a sylow $p$-subgroup of $G$ and $H \lhd G$ and $S\subset H$, how to show that if $S$ is normal in $H$ then its normal in $G$?

Let $G$ be some finite group. If $S$ is a sylow $p$-subgroup of $G$ and $H$ normal subgroup of $G$ and $S\subset H$, how to show that if $S$ is normal in $H$ then its normal in $G$?
2
votes
1answer
40 views

What is the difference between a $p$-group and a Sylow group?

We have just started a course about Group theory. I am confused about the difference between a $p$-group and a Sylow group. As I understand it, a group $G$ is called $p$-group if $|G|=p^{m}$, where $...
0
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0answers
27 views

Sylow p-subgroups of a finite group G which are contained in some other subgroup H are also Sylow p-subgroups in H

I have found a proof to the following question but I don't quite understand it. Let $P \leq H \leq G$ and $|G|=p^\alpha m$, where $p$ doesn't divide $m$. I need to show that if $P \in Syl_p(G)$, ...
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0answers
34 views

Representation of a finite group and its Sylow $p$-subgroup

Let $G$ be a finite group with order $|G|=p^n \cdot m$ for some positive integers $n,m$ and $H$ be a Sylow $p$-subgroup of $G$. What relations can we say about the representations of $G$ and $H$?
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1answer
52 views

There is no simple group of order $1452$.

I want to prove that there is no simple group of order $1452$. We have $1452 = 2^2\cdot 3\cdot 11^2$, and the Sylow theorems give: \begin{align} n_2 &\in \{1,3,11,33,121,363\} \\ n_3 &\in \{...
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0answers
21 views

G is cyclic if it is a finite p-group and has only one maximal subgroup [duplicate]

How can I show that for a finite $p$-group $G$,$G$ is cyclic if it has just one non trivial maximal subgroup?
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0answers
35 views

Find some subgroups of a group with 36 elements

Let a group $G$ with $\left | G \right | = 36$, then $G$ contains a normal subgroup of order $9$, $18$ or $3$. I can see groups of order $9$ in the case $n_{3} = 1$, by Silow's Theorem, but I do not ...
3
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2answers
47 views

$G$ contains a normal $p$-Sylow subgroup and $p$ divides the order of the center

I am looking at the following: Let $G$ be a non-abelian finite group with center $|Z|>1$. I want to show that if $G/Z$ is a $p$-group, for some prime $p$, then $G$ contains a normal $p$-Sylow ...
0
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2answers
31 views

Analogue of Third Sylow Theorem for sets

Let $n=p^em$ such that $p^e$ is the largest power of $p$ that divides $n$, and $p\nmid m$. Let $N$ be the number of subsets of order $p^e$ in a set $S$ with $|S|=n$. I want to compute $N\mod p$. Does ...
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1answer
25 views

How to prove a group of order $144$ is not simple using **Normalizers of Sylow intersections**.

How to prove a group of order $144$ is not simple using Normalizers of Sylow intersections. Here's what I have tried, but I am unable to proceed further. And if I proceed with Sylow-$2$ subgroups ...
2
votes
0answers
20 views

Show that if $\sigma=(a_1,a_2,\dots,a_m)$ and $\tau$ is any element of $S_n$, then $\tau\sigma\tau^{-1}=(\tau a_1,\tau a_2,\dots,\tau a_m)$ [duplicate]

Show that if $\sigma=(a_1,a_2,\dots,a_m)$ and $\tau$ is any element of $S_n$, then $\tau\sigma\tau^{-1}=(\tau a_1,\tau a_2,\dots,\tau a_m)$. I'm not quite sure how to start this. The solution starts ...
1
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1answer
60 views

Sylow subgroup of some factor group.

Let $G$ be a finite group. Let $K$ be a subgroup of $G$ and let $N$ be a normal subgroup of $G$. Let $P$ be a Sylow $p$-subgroup of $K$. Is $PN/N$ is a Sylow $p$-subgroup of $KN/N$? Here is what I ...
1
vote
1answer
23 views

Proof of a group of order $60$ is simple containing sylow $5$ subgroups.

So I was reading Sylow theorems from Dummit and Foote while I came across the following result I've understood the whole proof except the highlighted portion in red circle. Why $|H|\geq 1+4\times 6$? ...