# 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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### Let G be a group of order p^n suppose that H is a normal subgroup of G [closed]

Prove that there exists a normal subgroup H' a subset of G such that [H':H] = p
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### Product $PN$ of normal subgroups is abelian

I am trying to show that every non-abelian group $G$ of order $6$ has a non-normal subgroup of order $2$ using Sylow theory. First, Sylow's Theorem says the number of Sylow $2$-subgroups $n_2$ is ...
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### Upper-triangular matrices as union of centralizers of elements

Let $p$ be a prime number and $G=GL_n ( \mathbb{Z} / p \mathbb{Z} )$ such that $n\leq p$. Consider the set $U$ of upper-triangular matrices of $G$ having entries of $1$ on the diagonal. The ...
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### Let $|G|=2^np$, $p$ an odd prime, $H\unlhd G$ a Sylow $2$-subgroup with $H\cong(\Bbb{Z}/2\Bbb{Z})^n$, $p\nmid 2^n-1$. Prove $Z(G)$ is nontrivial.

Let $G$ be a group with $|G|=2^np$ ($p$ an odd prime). Let $H$ be a normal Sylow $2$-subgroup such that $H\cong(\mathbb{Z}/2\mathbb{Z})^n$. Prove that if $p$ does not divide $2^n-1$, then $G$ has a ...
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### Group of order $q^3p^3$, where $p,q$ are twin primes greater than $10$, is solvable

Let $q>p>10$ be twin primes, i.e., $q=p+2$. Show that every group of order $q^3p^3$ is solvable. This should be proven without using Burnside's theorem. Looking at the Sylow $p$-subgroup and ...
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### Let $G$ be a non-nilpotent group where all the non-normal abelian subgroups of $G$ are cyclic. Then $G$ has cyclic center.

Theorem : Let $G$ be a non-nilpotent group such that all the non-normal abelian subgroups of $G$ are cyclic. Then $G$ has cyclic center. Proof. Suppose that $Z(G)$ is non-cyclic. since $G$ is non-...
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### About finite group such that $Z(G)$ must be cyclic

Let $G$ be finite group and $G$ has at least one sylow subgroup $K$ such that $K\ntriangleleft G$ . proof if every nonnormal and abelian subgroup of G be cyclic then $Z(G)$ is cyclic. Theorem 1:...
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### Every subgroup of order 60 contains a subgroup of index 5

Aluffi IV.2.25 (in the chapter on Sylow theorems) suggests the following exercise: Assume $G$ is a simple group of order $60$. Use Sylow's theorems and simple numerology to prove that $G$ ...
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### the subgroups of special linear group

Let $G={\rm PSL}(2,8)$. So $|G|=2^3 \times 3^2 \times 7$. Let $P \in {\rm Syl}_2(G)$, $Q \in {\rm Syl}_3(G)$ and $R \in {\rm Syl}_7(G)$. I know $P \cong C_2 \times C_2 \times C_2$ and $Q,R$ are ...
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### Groups of order $pq$, $p$ and $q$ primes with $p<q$.

RIGHT ANSWER BY DUMMIT: Here Suppose $|G|=pq$ for primes $p$ and $q$ with $p<q$. Let $P \in Syl_{p}(G)$ and let $Q \in Syl_{q}(G)$. We are going to show that $Q$ is normal in $G$ and if $P$ also ...
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### Invariant Sylow subgroups

Today, I'm reading lemma 2.2.c of an article by Antonio Beltran. Lemma 2.2. Suppose that A is a finite group acting coprimely on a finite group G, and let $C = C_G(A)$. Then, for every prime p, ...
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### Fixed points of the conjugation action of a p-Sylow subgroup on the set of p-Sylow subgroups

Aluffi IV.2.10 suggests the following exercise. Let $P$ be a $p$-Sylow subgroup of a finite group $G$, and act with $P$ by conjugation on the set of $p$-Sylow subgroups of $G$. Show that $P$ is the ...
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### Prove that $K$ is a Sylow $p$-subgroup of $T$.

Let $G$ be a finite group and let $T$ be a subgroup of $G.$ Suppose $K$ is a Sylow $p$-subgroup of $G$ such that $K \subseteq T.$ Since $K$ is a Sylow $p$-subgroup of $G$, then $|K|=p^m.$ I want to ...
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### Decomposing the alternating group $A_5$ as “product” of its Sylow subgroups

My professor remarked in class today that the alternating group $A_5$ can be decomposed as a product of its Sylow subgroups but only if we align the Sylows in a specific way and combine them in the ...
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### Understanding a Lang's proof of Sylow theorem (related to the class formula)

This proof comes from S. Lang's Algebra Revised Third Edition. Honestly 2/3 of this proof makes sense to me, except the application of class formula and the followings. My questions are: Why do we ...
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### Prove that there is no simple group of order $216=2^{3}\cdot 3^{3}$. [duplicate]

Prove that there is no simple group of order $216=2^{3}\cdot 3^{3}$. I feel like I am supposed to use the Index Theorem here but when I use Sylow's Third Theorem I have that $n_{3}\in\{1,4\}$. I am ...
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### Frattini sugroup and normal subgroup

For any group $G$, let $\Phi(G)$ denote the Frattini subgroup of $G$. Let $G$ be a finite group, such that $\dfrac{G}{\Phi(G) } \cong K \times \mathbb{Z}_{p}$, where $p$ is prime number. Prove that ...
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### Proving $P$ is a Sylow $p$-group of $PN$

I am having trouble solving the following problem: Let $G$ be a finite group of order $p^an$, where $p$ is a prime and $p \nmid n$. Let $P$ be a Sylow $p$-group in $G$ and let $N \unlhd G$. It can ...
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### Show that a group of order 12 cannot have nine elements of order 2.

This is what I have so far. Assume the contrary, then by Sylow's Third Theorem, the number of Sylow 3-subgroups is either 1 or 3. If it were 1 then there are only 7 elements of order 2. Thus there are ...
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### What are the possibilities for the number of elements of order $5$ in a group of order $100$?

I want to make sure that I am doing this right. Since $100 = 5\cdot 20$, by Sylow's Third Theorem we have that $n_{5}\equiv 1$ (mod 5) and $n_{5}\mid 20$. So $n_{5}\in\{1,6,11,16,21,26,\dots\}$ and ...
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Here : https://groupprops.subwiki.org/wiki/A5_is_the_unique_simple_non-abelian_group_of_smallest_order it is shown that the only simple subgroup of order $\ 60\$ upto isomorphy is $A_5$ I do not ...
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### Cardinality of direct product of Sylow $p$-subgroups

Let $G$ be a finite group such that, for all prime number $p$, $P_p$ is a normal Sylow $p$-subgroup of $G$. Let $I$ denote the set of prime numbers dividing $|G|$ and K=\bigcup_{n\in\mathbb{N}}\{g\ |...
### Quotient of quotient groups and Sylow $p$-subgroups
Let $G$ be a finite group and $N$ a normal subgroup of $G$. Let $\pi:G\rightarrow G/N$ be the canonical homomorphism. Suppose $P$ is a Sylow $p$-subgroup of $G$. Then $\pi[P]\leq G/N$. Note that $G$ ...