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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$G$ be a finite group and $f \in Aut (G)$ such that $f^3$ is identity and $f$ has unique fixed point , then any $p$-Sylow subgroup is normal?

Let $G$ be a finite group and $f \in Aut (G)$ such that $f^3$ is identity and $f(x)=x \implies x=e$ ; then is it true that for every prime $p$ dividing $|G|$ , there is exactly one $p$-Sylow subgroup ...
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18 views

Sylow counting argument; prove G isomorphic to the direct product.

Let G be a group of order $|G|=pq^m$, where $p$ and $q$ are primes with $q^m<p$. i) Use a Sylow counting argument to show that $G\cong C_p\rtimes_hQ$ where Q is a group with $|Q|=q^m$ and ...
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1answer
12 views

Calculating $| \langle \cup_{i=1}^n P_i \rangle |$ where $P_i$ are Sylow subgroups of G

I'm trying to prove: Let $\lbrace P_i: i \in I \rbrace$ be a set of Sylow subgroups of a finite group G, one for each prime divisor of $|G|$. Then $\langle \cup_{i \in I} P_i \rangle = G$. (From ...
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1answer
40 views

Normalizers of subgroups of Sylow $p$-subgroups

I am wondering whether there is an easy example of a finite group $G$ with a Sylow $p$-subgroup $P$ and a subgroup $Q\leq P$ such that the normalizer $N_P(Q)$ of $Q$ in $P$ is NOT a Sylow $p$-subgroup ...
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1answer
40 views

Prove that $g\in P$

Let $G$ be a finite group and $P$ a Sylow $2$-subgroup. If $g$ is an element of $G$ with order a power of $2$ then it lies in some conjugate of $P$. I get this. But if also $g\in C_G(P)$, then it is ...
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2answers
34 views

$P$ is a Sylow subgroup of $G$ then $P$ is normal in $G$.

Let $G$ be a finite group with subgroups $H$ and $P$ and if $H$ is normal in $G$ and $P$ is normal in $H$ and $P$ is a Sylow subgroup of $G$ then $P$ is normal in $G$. Is the statement true, I heard ...
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52 views

Sylow subgroups of $\text{SL}_2(q)$.

Let $p,q$ be primes such that $p$ is a divisor of $|\text{SL}_2(q)|=(q-1)q(q+1)$. Hence $\text{SL}_2(q)$ admits non-trivial Sylow subgroups. I am interested in the isomorphism type of the $p$-Sylow. ...
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68 views

Classify all groups of order $p^2q^2$ up to isomorphism

Let $p,q \in \mathbb{N}$ be prime numbers with the properties $2 < p < q$ and $q - 1 , q + 1 \notin \left\langle p \right\rangle$ Classify all groups of the order $p^2q^2$ up to isomorphism. ...
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3answers
41 views

Prove that G has a normal Sylow p-subgroup

Suppose that $|G| = pq$ where $p$ and $q$ are distinct primes such that $p$ does not divide $q-1$. Prove that G has a normal Sylow $p$-subgroup . I know what by Sylow's Theorem, either $n_p=1$ or ...
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59 views

Order of an Equivalence Class and the number of Coset.

Let $ \mathcal{M} $ be the set of all subsets of finite group $G$ which have $p^{\alpha} $elements. Thus $ \mathcal{M} $ has $ {p^{\alpha}m \choose p^{\alpha}} $ elements. Given $M_1 ,M_2 \in ...
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1answer
22 views

Inverse image of Sylow $p$-subgroup is Sylow $p$-subgroup

I am trying to prove or disprove the following statement: Let $H$ and $G$ be finite groups, $p$ a prime number, $\psi: H \to G$ an injective homomorphism and $S$ a Sylow $p$-subgroup of $G$ with ...
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2answers
46 views

Conjugates of a sylow 2-subgroup of $A_5$

The exact problem is as follows: Show that a Sylow 2-subgroup of $A_5$ has exactly 5 conjugates. My question is why are there not 15 conjugates. Seeing as every Sylow 2-subgroup of $A_5$ is the ...
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1answer
46 views

Can a Simple Group possess this property? [closed]

If a simple group G is of order 168 then can I find subgroup of order 7 of G ? If so, then what is the number of subgroups of G of order 7 ?
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1answer
39 views

Prove a finite group has non-normal Sylow p-subgroup of order p if $n_{p}\neq 1\pmod{p^2}$

Let $G$ be a finite group and p a prime that divides $|G|$. Let $n_{p}$ denote the number of Sylow p-subgroups of $G$. Prove that if $n_{p}\neq 1\pmod{p^{2}}$, then for any Sylow p-subgroup $P_{1}$ of ...
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1answer
42 views

show that a Sylow p-group lies in the center of $G$

I'm stuck with the following problem: Let $G$ be a finite group and $p$ be the smallest prime dividing $|G|$. Suppose the Sylow p-subgroup $H$ of $G$ is normal and cyclic. Show that $H$ lies in the ...
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3answers
60 views

If $ A \trianglelefteq G $, and $ B \trianglelefteq A $ is a Sylow subgroup of $ A $, then is $ B \trianglelefteq G $?

Let $ A \trianglelefteq G $ and $ B \trianglelefteq A $ a Sylow normal subgroup of $ A $. My textbook says then that $ B \trianglelefteq G $. I don’t understand why that is.
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4answers
101 views

Is there a non abelian group of order 759? [closed]

I tried to use Sylow theorems to prove that there is not, but it is not trivial.
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2answers
53 views

characterizing groups of order 1960

This problem is from an an Introduction to Abstract Algebra by Derek John that I am solving. I am trying to prove that any group of order 1960 aren't simple, so I am doing it by contradiction, but I ...
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1answer
96 views

Finding and classifying all groups of order 12

I was working on classifying all the groups of order 12. I dug around at some of the previous questions here and while they address the idea, none of them were entirely satisfactory: Classifying ...
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0answers
30 views

Show that there does not exist a simple group of order $120$. [duplicate]

Show that there does not exist a simple group of order $120$. By the Sylow's theorem, I already know that $N_5 | 24$ and $N_5 \equiv 1 \pmod 5$; I found that $N_5 \in \{1,6\}$ I think I can use ...
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48 views

Show that there does not exist a simple group of order $126$.

Show that there does not exist a simple group of order $126$. By the Sylow's theorem, I already know that $N_7 | 24$ and $N_7 \equiv 1 \pmod 7$; I found that $N_7 \in \{1,8\}$ I think I can use the ...
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Number of elements of order $5$ in group of order $5\times 13\times 43\times 73$

Let $G$ be a group of order $5\times 13\times 43\times 73$. Find the number of elements of order $5$. Here is what I do: Since $|G| = 5m$ where $(m,5) = 1$, $m = 13\times 43\times 73$, by Sylow's ...
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26 views

Size of the intersection of a $p$-Sylow subgroup and a normal subgroup.

Assume $G$ is a group whose size is $(p^r)*m$, where $p$ is prime and $p$ doesn't divide $m$. Let $P$ be a $P$-Sylow subgroup of $G$, and $H$ a normal subgroup of $G$. Lagrange's theorem gives us that ...
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2answers
113 views

Group of order $255$ is cyclic

Let $G$ a group and its order is $255$. Prove that $G$ is cyclic. I easily demonstrated that the group has only one $17$-Sylow subgroup $P$ that is normal in $G$ and it's cyclic since it is of ...
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1answer
29 views

Let $G$ be an Abelian group with $|G| = n$. Let $p$ be prime with $p | n$. Show that the Sylow p-subgroup of $G$ consists of $e$ and ..

Let $G$ be an Abelian group with $|G| = n$ and let $p$ be prime with $p | n$. Show that the Sylow p-subgroup of $G$ consists of $e$ and all elements whose order is a power of $p$. Answer: By ...
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In $S_{4}$, find a Sylow 2-subgroup and a Sylow 3-subgroup.

Q: In $S_{4}$, find a Sylow 2-subgroup and a Sylow 3-subgroup. A: With everyone's comments and inputs, I have outlined the following answer. Thank you all for the guidance. $|S_{4}|= 24 = ...
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1answer
48 views

Classify the group of order 33, sylow theorem

From artin's 2nd edition of Algebra on the sylow theorem section. Problem: Classify groups of order (a) 33 Answer: Let $G$ be the group. $n(G)=33$ Factors of $33$ are $1,3,11,33$ If for any any ...
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3answers
192 views

Different Applications of Sylow Theorems

The theorems of Sylow are very well known and almost every mathematician learns in his undergraduate course. The applications of Sylow theorems given in books are of the kind "If $|G|=....$ then ...
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1answer
42 views

Solution check: the number of elements of order $p$ and Sylow $p$-subgroups in $S_p$?

This is my original problem: Find the number of elements of order $7$ in $S_7$. Find the number of Sylow $7$-subgroups of $S_7$. Since $7$ is prime, we know that there must be a $7$-cycle in ...
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1answer
48 views

Each group of order $p^2 q$ possesses a proper normal subgroup, where $p$,$q$ are primes.

Each group of order $p^2 q$ possesses a proper normal subgroup, where $p$,$q$ are primes. A solution of this question is given here: You can use a counting argument. Note that if there are ...
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1answer
41 views

If $C_G(x) \leq H$ for every $p$-element $x \in H$, then $p$ cannot divide both $|H|$ and $|G:H|$

This is problem 1.D.2 in Isaacs, Finite Group Theory. I am self-studying, so would appreciate a proof verification. Note: in this book, all groups are assumed finite unless otherwise stated. Fix ...
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1answer
35 views

If $P \in Syl_p(G)$ and $P$ is cyclic, then $N_G(P)=C_G(P)$

Let $G$ be a group such that $|G|=p^a m$ where $p$ is the smallest prime divisor of $|G|$. If $P \in Syl_p(G)$ and $P$ is cyclic, then $N_G(P)=C_G(P)$ Proof First, note that $C_G(P) \leq ...
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46 views

$\pi$-complement of direct product of groups

Let $G = N_1 \times N_2 \times \ldots \times N_n$ be a finite group and the direct product of the normal subgroups $N_i$. Let $\pi$ be a set of primes, then a $\pi$-complement is a subgroup $K$ such ...
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1answer
37 views

Center of normalizer of cyclic group

Given a Group of order $pm$ with $p$ the least prime dividing the order of $G$ and p does not divide m. Is it true that a $p$-sylow subgroup is contained in the center of its normalizer. I think yes, ...
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Order of the normalizer of Sylow-$p$ subgroup of $A_{p+1}$.

Let $P \in Syl_{p}(A_{p+1})$, where $A_{p+1}$ is the alternating group on $p+1$ elements. Show that $N_{A_{p+1}}(P)$ has order $p(p-1)/2$. Attempt: I'm supposed to consider all elements of order $p$. ...
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1answer
40 views

Number of Non-Abelian Groups of order 21

My goal is to count the number of non-abelian groups of order 21, up to isomorphism. I also need to show their presentations. This is a homework assignment, so I would appreciate leads rather than ...
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99 views

There is no simple group of order 56

I'm working on question 16 of http://www.austinmohr.com/Work_files/hw4.pdf I'm confused by why there are 48 elements of order 7. I understand that $n_7 = 8$, but no further.
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60 views

Group of Order 105 Having Normal Sylow 5/7 Subgroups

I am trying to prove that, given $|G|=105$, the G has a normal Sylow 5 subgroup and a normal Sylow 7 subgroup. I think the thing that is confusing me is the word "and". It would seem that there ...
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On $p$-subgroup of the general linear group over a vector space over $\mathbb{Z}_p$

Suppose $V$ is a finite-dimensional vector space over $\mathbb{Z}_p$ and $G$ is a $p$-subgroup of $GL(V)$. Then there exists such non-zero vector $v\in V$ that $gv=v$ for all $g\in G$. Moreover, we ...
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1answer
41 views

Normality of Sylow $p$-subgroup if all maximal subgroups have prime index

This is exercise 1.C.7 in Isaacs, Finite Group Theory. I have a solution, but I am suspicious that there is something wrong with it, because I do not fully use one of the hypotheses. Let $G$ be a ...
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1answer
57 views

If $|G| = 120$ then $G$ has a subgroup of index $3$ or $5$ (or both)

This is exercise 1.C.4 in Isaacs, Finite Group Theory. I think I have a proof, but would like to verify the proof and also inquire whether it can be shortened or improved significantly. Let $|G| = ...
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2answers
53 views

Showing $|G|=90$ means $G$ is not simple by centraliser argument

I want to show that a group $G$, with $|G|=90$ cannot be simple, specifically using a centraliser argument. The exercise gives a walkthrough really of what I am to do, but I am even then, still having ...
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84 views

Showing $\left\lvert G\right\rvert=105$ is not simple. Proof verification

I want to know if my proof showing that a group, $G$, where $\left\lvert G\right\rvert=105$ cannot be simple, is correct. $\left\lvert G\right\rvert=105=3\times5\times7$ gives us: ...
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Reasoning about subgroups $H \le PSL(2,q)$ based on knowledge about subgroups of normalizers in it

Suppose that $G$ is a finite permutation group acting transitively and non-regularly on $\Omega$. Also suppose that each non-trivial element has at most two fixed points and $|\Omega| \ge 4$. Let me ...
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16 views

On the image of $\pi$-Hall subgroup in $G / G'$.

Let $G$ be a finite group and $\pi$ a non-empty set of primes and let $P$ be a $\pi$-Hall subgroup of $G$. Then we have $PG' / G' = O_{\pi}(G/G')$, and so as $G/G'$ is abelian $$ G / G' = PG'/G' ...
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If all Sylow subgroups of $G$ intersect in Sylow subgroups of solvable subgroup $H$, then $H$ is subnormal

Let $H \le G$ be a solvable subgroup of the finite group $G$ such that for each prime $p$ and each Sylow $p$-subgroup $S$ of $G$ we have $$ S \cap H \in \mbox{Syl}_p(H). $$ Then $H$ is ...
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1answer
26 views

$\vert PSL(4,2) \vert = \vert PSL(3,4) \vert $ but $PSL(4,2) \ncong PSL(3,4)$.

Prove that $\vert PSL(4,2) \vert = \vert PSL(3,4) \vert $ but $PSL(4,2) \ncong PSL(3,4)$. Attempt: I have shown that the orders of the two groups are equal. Then, consider the set of unitriangular ...
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1answer
32 views

The Sylow 2-subgroup of $SL(2,3)$.

Why is the Sylow 2-subgroup of $SL(2,3)$ normal? I know that $n_2 \in \{1,3\} $, where $n_2$ is the number of Sylow 2-subgroups. But how do I show that $n_2 \neq 3$?
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1answer
23 views

Find the number of Sylow $p$-subgroups of $G$.

Let $G=GL(2,\mathbb Z_p)$. Find a Sylow $p$-subgroup of $G$ and find the number of Sylow $p$-subgroups of $G$. My try: $|G|=(p^2-1)(p^2-p)=p(p-1)^2(p+1)$. Hence any subgroup of order $p$ is a Sylow ...
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1answer
87 views

Question about proof that group of order 48 must have a normal subgroup of order 8 or 16

Prove a group of order $48$ must have a normal subgroup of order $8$ or $16$. Solution: The number of Sylow $2$-subgroups is $1$ or $3$. In the first case, there is one Sylow $2-$subgroup of ...