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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
84 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,...
3
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1answer
67 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
37 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 = ...
3
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0answers
69 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
55 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
70 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
46 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
30 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
37 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 ...
3
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2answers
38 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
72 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 ...
4
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1answer
133 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
43 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
117 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-...
1
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1answer
30 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
46 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
39 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
76 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
73 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 $G$ ...
5
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2answers
74 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
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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 $|P|=2^5$. ...
0
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1answer
41 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
96 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
23 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
79 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
44 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
50 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
41 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
77 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
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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 ...
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1answer
25 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 ...
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0answers
31 views

Number of proper nontrivial subgroups for a group $G$ with size $pq$?

If I have a group $G$ with order $pq$ with $p,q$ primes, $p\neq q$. Then I want to try to apply Sylow's third theorem. I want to argue that there is only 2 proper subgroups of $G$. Since $p$ ...
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1answer
31 views

G has 4-Sylow 3-subgroups

I am working on the following problem: Prove that if $\lvert G \rvert = 12$ and $G$ has $4$ Sylow 3-subgroups, then $G \equiv A_4$. If you let $G$ acts by conjugation on the set containing the $4$ ...
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2answers
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Group Theory Sylow Subgroup [closed]

What's an example of a group $G$ and an integer $n$ dividing $|G|$ with $0 < n < |G|$ such that $G$ has no subgroup of order $n$.
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1answer
64 views

Every Sylow subgroup is normal, then $G$ has a subgroup of order $m$ for every division $m$ of $|G|$

Some trouble working out an algebra problem. Suppose that every Sylow subgroup of a finite group $G$ is normal. Prove that $G$ has a subgroup of order $m$ for every divisor $m$ of $|G|$.
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No group of order 10000 is simple

A proof of this fact was already given here: No group of order 10,000 is simple However, I am wondering whether or not the following proof works as well: By way of contradiction, suppose $G$ is ...
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0answers
30 views

The cases in proving that a group of order 90 is not simple

I am trying to attempt this problem, but I am wondering why exactly these are the two cases the problem is split into. I can understand the first case, since that lets us count elements and get a ...
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1answer
43 views

Group Theory Subgroups Sylow Theory [closed]

Let $G$ be a finite group, $p$ a prime and $e$ a nonnegative integer. If $p^e$ divides the order $|G|$ of $G$, show that $G$ has a subgroup of order $p^e$.
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1answer
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Group of order greater than 8 doesn't decompose into a direct product and Sylow 2-subgroup isomorphic quaternion group

Is there a group of order greater than 8 that does not decompose into a direct product such that its Sylow 2-subgroup isomorphic quaternion group $Q_8$?
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2answers
39 views

Question about p-Sylow subgroups being maximal

I am having some trouble wrapping my head around p-Sylow subgroups at the moment. I am given that for p, prime, a p-Sylow subgroup of G is a maximal p-subgroup that is not a proper subgroup of any ...
2
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1answer
64 views

Group of order $6p^m$ is solvable for prime $p\geq 7$

Let $p\geq 7$ be a prime and $m$ be a positive integer. Prove that group of order $6p^m$ is solvable. Attempt: By Sylow's theorems we have that $n_p \mid 6$ so $n_p\in \{1,2,3,6\}$ where $n_p$ is ...
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1answer
74 views

Proof that no group of order $525$ is simple

I would like some verification that any group $G$ of order $|G| = 525 = 5^2 \cdot 3 \cdot 7$ is not simple. I've attached my argument below. Please let me know if you see any issues. Thanks Let $G$ ...
1
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1answer
57 views

Let H be a Sylow p-subgroup of G. Prove that H is the only Sylow p-subgroup of G contained in N(H).

Let H be a Sylow p-subgroup of G. Prove that H is the only Sylow p-subgroup of G contained in N(H). I saw a proof online that was pretty long, but can't I just argue that if $H \subset N(H)$, then $H$...
2
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2answers
44 views

Group Isomorphism regarding Sylow Subgroups

Suppose I have given two groups say, $G_1,G_2$ such that they have same order.I'm assuming they are non commutative.Then their Syllow subgroups has same order clearly.If I'm given that the number of ...
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0answers
42 views

$gcd(p,m-1)=gcd(p-1,m)=1$ if and only if G has a normal Sylow p-subgroup in $Z(G)$

Suppose that $|G|=pm$ where $p$ is a prime and $p\not|m$. Prove that $gcd(p-1,m)=gcd(p,m-1)=1$ iff G has a normal Sylow p-subgroup in $Z(G)$ I have a problem in both directions. How can I solve ...
2
votes
2answers
34 views

Is $n_p(G)$ unique for different groups of size $p$?

If $G$ has that $|G| = p^am$ where $p$ is prime and $gcd(p,m) = 1$, we have that $n_p(G)$ counts of Sylow $p$-subgroups in $G$. If $P$ is a Sylow $p$-subgroup, by the third Sylow theorem \begin{align*...
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0answers
36 views

Group Theory (Sylow p-subgroups questions)

Let $GL_n\mathbb{F}_p$ be the group of invertible $n \times n$ matrices with entries coming from $\mathbb{F_p}=\{0, 1, ..., p-1\}$ and with group operation multiplication of matrices. (We are writing $...
0
votes
1answer
29 views

Sylow's First Theorem acting on Abelian Group

Background In the book of Judson's book on abstract algebra, Sylow's First Theorem is proved by first invoking the class equation and then considering the case where $p$ can/cannot divide $[G:C_G(g)]$...
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1answer
56 views

Equivalent conditions for a Group $G$ with order $p^2q$ ( with $p>q$ both prime) be abelian.

I saw this homework many times, but always asks in the statament that $p^2 \not\equiv 1$ (mod $q$) and $q \not\equiv 1$ (mod $p$). But today in a book text of Galois theory I Saw a similar example ...