# 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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### Show that $H$ is a normal subgroup of $G.$

This question was asked in my mock test of masters entrance test and I couldn't prove one of the question: Question $\to$ Let $G$ be a group of order $105$ and $H$ be it's subgroup of order $35$. ...
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### Let $G$ be a group of order $p^nq$ where $p$ and $q$ are distinct primes and suppose $q \nmid p^i-1$ for $1 \leq i \leq n-1$. Prove $G$ is solvable

This is an extension of this post. Let $G$ be a group of order $p^nq$ where $p$ and $q$ are distinct primes and suppose $q \nmid p^i-1$ for $1 \leq i \leq n-1$. Prove that $G$ is solvable. This can be ...
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### If $|G| = p^a q^b$ for $p$, $q$ prime, and $G$ has exactly one Sylow-$p$ subgroup $P$ and one Sylow-$q$ subgroup $Q$, then $G \cong P \times Q$

I am hoping to show that if $|G| = p^a q^b$ for $p$, $q$ prime, and $G$ has exactly one Sylow-$p$ subgroup $P$ and one Sylow-$q$ subgroup $Q,$ then $G \cong P \times Q$. Here's where I've gotten so ...
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### Generalized Sylow's theorem [closed]

I'm working through some exercises in Alperin and Bell's textbook "Groups and Representations." I came across a very interesting exercise which generalizes Sylow's theorem: Ex 7.4: If $|G|$ ...
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### Is restriction to $p$ Sylow subgroup $\text{res} : H^1(H,M)\to H^1(H_p,M)$ injective?

'Galois cohomology of Algebraic number fields' written by K. Haberland reads the following lemma in page 66. Let $H$ be a finite group. Let $p$ be a prime number and $H_p$ be a fixed p Sylow subgroup ...
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### Normalizer of an $A$-invariant Sylow $p$-subgroup

I was reading the Antionio Beltrán and Changguo Shao article On the number of invariant Sylow subgroups under coprime action and there is a part of the Lemma 2.5. which I do not undertand. First of ...
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### On the number of invariant Sylow subgroups under coprime action - Antonio Beltrán and Changguo Shao article

This is an article that Antonio Beltrán and Changguo Shao wrote. Lemma 2.5. states: [All groups are supposed to be finite (this is mentioned before)] Lemma 2.5. Let $A$ be a group acting coprimely on ...
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### Standard representation restricted to $S_5$ of $S_6$ acting on its set of sylow $5$-subgroups.

This is from a practice exam (paraphrasing): Let $\chi$ be a character of $S_6$, given by $\chi(g) = |\text{Fix}(g)|-1$ for the action of $S_6$ on the set of its sylow $5$-subgroups. Decompose the ...
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### Show that there is no simple group of order 351

I want to show that there is no simple group of order $351 = 3^3 \cdot 13$. Let $G$ be a group. of order 351. Using Sylow III, I found that $n_3 \in \{1,13\}, \quad n_{13}\in \{1,27\}$. Let us suppose ...
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### Let $P \in \text{Syl}_{5}(S_5)$. Show that the normalizer $N := N_{S_5}(P)$ is a monomial group.

Fix the field to $\mathbb{C}$. Let $P$ be a sylow $5$-subgroup of the symmetric group $S_5$. Let $N := N_{S_5}(P)$ be the normalizer of $P$. I want to show that $N$ is a monomial group, that is, that ...
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### Show that every group of order $2 · 3 · 5 · 67$ has a normal subgroup of order $5$. [duplicate]

Show that if $G$ is a group of order $2010 = 2 · 3 · 5 · 67$, then $G$ has a normal subgroup of order $5$. I tried use Sylow's theorem to show that $k_5=1$. $k_5$ is equal to $1$, $6$, or $201$.
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### Proving that a group of order 360 has 10 Sylow 3 -subgroups ，36 Sylow 5 -subgroups，45 Sylow 2 -subgroups is a simple group

Edit: This is what the OP is trying to ask, I think. Let $G$ be a group of order $360$. Suppose that there are ten Sylow $3$-subgroups, 36 Sylow $5$-subgroups and 45 Sylow $2$-subgroups. Show that ...
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### If a p-group is normal then can we say G is going to be a direct product of P

I recently came across a theorem that states that if G is a finite group and every p-group of G is normal then G is isomorphic to the direct product of its Sylow p subgroups. To prove this we use that ...
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### Deducing there exists exactly $5$ isomorphism classes of groups of order $12$.

There's a substantial amount that's been written about the semi-direct products of a group of order $12$ on this website. However, there's something that seems to be taken for granted each time the ...
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### Groups of order $2^n p$ for $n\geq 1$ and $p$ prime with $2^n> (p-1)!$ are non-simple. Is my proof correct?

I'm doing my homework in Group Theory and as part of an exercise, I want to show the following Lemma: Let $n\geq 1$, $p$ a prime, s.t. $2^n > (p-1)!$ and $G$ a group of order $2^n p$. Then G has a ...
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### G has a element of order 2 not lying in center

Let G be order of 8，if G has a element of order 2 not lying center，how to prove G is isomorphic to $D_8$? A hint is consider the sylow-2 subgroup of $S_4$, I know it's $D_8$. So I want to construct a ...
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### Center of group of order 3773

What can we say about $|Z(G)|$ if $G$ is of order $3773 = 7^3 * 11$. Here $|Z(G)|$ means the size of the center. This is an exercise in book about Sylow theorem and I have no idea what Sylow theorem ...
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### Prove the index of a proper subgroup of a simple group of order 17971200 is at least 14.

I didn't find a solution for this problem or other usual approaches that could directly work. So, here is my attempt. I am self-studying and reviewing group theory recently, and would like to know if ...
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### Orbits of same size under conjugation action of Sylow $p$-subgroups by a normal subgroup $H$

Let $H$ be a normal subgroup of a finite group $G$ and $p$ a prime number, show that the orbits of the conjugation action induced by $H$ on $X = \{P \leq G \mid P \in Syl_p(G) \}$ all have the same ...
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### Order of a Sylow $p$-subgroup of the symmetric group of order $p!$ [closed]

I have been struggling with this question because many times I saw the statement "order of Sylow $p$-subgroup of symmetric group of order $p!$ is $p$" being used in exercises and texts ...
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### Are the character tables for $Z$-groups known?

A $Z$-group is a group whose Sylow subgroups are all cyclic groups. I know from here that if two $Z$-groups have the same character table, then they are in fact isomorphic groups. To my understanding, ...
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### Groups of order $231$

In a certain exercise am I asked to show that a group of order $231$ has always an abelian subgroup of order $33$. It might be simple, but I still get struct at the same point. Here is my approach: ...
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### On p-Sylow subgroups of locally finite groups

I've come across this problem (I'm sorry if it's poorly written but it's a translation); for a $p$-Sylow I mean a $p$-subgroup that is not contained in any other $p$-subgroup: Let $G$ be a locally ...
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### Every group of order p^2 is abelian. [closed]

I am studying for my algebra qualifying exam in January, and I have a question regarding the proof I have for this question in Hungerford, problem 13 under the Sylow Theorems section. The way I ...
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### Supposed contradiction from Sylow theorems

Let $D_{506}$ denote the dihedral group with $1012 = 4 \cdot 23 \cdot 11$ elements. The Sylow theorems tell us that the number of 2-Sylow-subgroups (that is, subgroups of order 4) of $D_{506}$ divides ...
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