Articles of sylow theory

P p-sylow with $ P ⊂ Z(G) $

For me this problem is hard . If $ P⊂ Z(G)$ is a $p$-sylow of $G$ then there is a $N$ normal subgroup of $G$ such that $P ∩ N = 1$ and $G = PN$. I try use the Schur-Zassenhaus: All normal subgroup cop rime has a complement and all conjugates are conjugate, but […]

The intersection of two Sylow p-subgroups has the same order

Let $G$ be a finite group and assume it has more than one Sylow $p$-subgroup. It is known that order of intersection of two Sylow p-subgroups may change depending on the pairs of Sylow p-subgroups. I wonder whether there is a condition which guarantees that intersection of any two Sylow $p$-subgroups has the same order. […]

How to apply Thompson's A×B lemma to show this nice feature of characteristic p groups?

How does one apply Thompson’s A×B lemma (Lemma 24.2 on page 112 of Aschbacher’s Finite Group Theory) in order to prove this nice lemma (Lemma 31.16 on page 160)? In the book, I basically don’t understand what “G,” “A,” and “B” (from Thompson’s A×B lemma) should be in the second sentence of the proof of […]

Enumerating Sylow $2$-subgroups of Dihedral Group (of order $2^{\alpha}k$ for $k$ odd).

Let $2n = 2^a k$ for $k$ odd. Prove that the number of Sylow $2$-subgroups of $D_{2n}$ is $k$. I managed to prove this result by showing that the normalizer of any Sylow $2$-subgroup is itself. The result immediately follows, in that case. However, my problem is that I came across a different solution to […]

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 a normal subgroup of order $16$ so we are done. In the second case, let $G$ act by conjugation on the Sylow $2$-subgroups. […]

Prove that a non-abelian group of order $pq$ ($p<q$) has a nonnormal subgroup of index $q$

So I’ve come up with a proof for the following question, and I’d like to know if it’s correct (as I couldn’t find anything online along the lines of what I did). Question Let $p$ and $q$ be primes with $p<q$. Prove that a non-abelian group of order $pq$ has a nonnormal subgroup of index […]

The number of Sylow subgroups on $G$ with $|G|=pqr$

I’m doing a part of an exercise and I don’t know how to go on. Here it goes: Let $G$ be a group, with $|G|=pqr$, $p,q,r$ different primes, $q<r$, $r \not\equiv 1$ (mod $q$), $qr<p$. Show $G$ has an unique Sylow $p$-subgroup $P$. Also suppose that $p \not\equiv 1$ (mod $r$), $p \not\equiv 1$ (mod […]

How to show that $H \cap Z(G) \neq \{e\}$ when $H$ is a normal subgroup of $G$ with $\lvert H\rvert>1$

Let $G$ be a group of order $p^n$, $p$ a prime, $n>1$ and $H$ a normal subgroup of $G$ with $\lvert H\rvert>1$. Show that $H \cap Z(G) \neq \{e\}$.

Classifying groups of order 12.

I was trying to classify groups of order 12 and I ended up with 5 different groups: $\bullet$ $\Bbb{Z}_{12}$ $\bullet$ $\Bbb{Z}_2 \times \Bbb{Z}_6$ $\bullet$ $(\Bbb{Z}_2 \times \Bbb{Z}_2) \rtimes_{\alpha} \Bbb{Z}_3$ where $\alpha: (1,1) \rightarrow \bar{-1}$ $\bullet$ $(\Bbb{Z}_2 \times \Bbb{Z}_2) \times \Bbb{Z}_3$ $\bullet$ $\Bbb{Z}_3 \rtimes \Bbb{Z}_4$ where $\alpha$ sends the generator to $\bar{-1}$ I want to show […]

Classifying groups of order 21.

We have two sylow subgroups of orders 7 and 3. Let $n_3$ and $n_7$ denote the number of sylow subgroups for 3 and 7, respectively. $n_7 \equiv 1 \mod 7$ and $n_7 | 3 \implies n_7 = 1$ $n_3 \equiv 1 \mod 3$ and $n_3 | 7 \implies n_3 = 1, 7$ Let $P_3 \cong […]