Articles of group theory

Prove that any group $G$ of order $p^2$ is abelian, where $p$ is a prime number

Possible Duplicate: Showing non-cyclic group with $p^2$ elements is Abelian “Let $p$ be a prime number. Prove that any group $G$ of order $p^2$ is abelian. You may assume the fact that the centre of a $p$-group is non-trivial”. I understand from the question is that the group $G$ is a $p$-group, with $p^2$ number […]

A problem about normalizers in $PSL(V)$

Let $K=\mathbb F_{p^k}$ a finite field, and $V$ a vector space on $K$. Clearly $PSL(V)=SL(V)/SL(V)\cap Z$ acts on $V$ by the following rule ($Z$ is the subgroup of the scalar functions): $$v^{(SL(V)\cap Z)\gamma}:=\gamma(v)$$ this action is well defined, and if $W$ is a proper subspace of $V$ $$N_{PSL(V)}(W):=\{x\in PSL(V)\,:\, w^x\in W\quad\forall w\in W\}$$ Such normalizers […]

Non-commutative quotient group?

If you have a non-abelian group $G$ with some normal subgroup $K$, is it possible to have a non-abelian quotient group $G/K$? Besides actually sitting down and trying to generate quotient groups through exhaustion, I have been thinking about using the fundamental theory of homomorphisms to pick a small non-abelian group like $D_6$ and find […]

The kernel of an action on the orbits of normal subgroup if group acts such that $\mbox{fix}(g) \in \{0,p\}$ for $g \ne 1$

Let $G$ be a permutation group acting transitively on $\Omega$ and suppose $N \unlhd G$ is a normal subgroup of $G$. Assume that for $g \in N_g(G_{\alpha})$ we have $$ G_{\alpha} \cap G_{\alpha}^g = 1 $$ for some point stabilizer $G_{\alpha}$. Also assume that every nontrivial element either has no fixed point or exactly $p$ […]

Formal derivatives over finite fields.

I am slightly confused about what formal derivatives over finite fields mean. Example 1: Consider $f(x)=x^3-2\in \mathbb{F}_7[x]$. By checking each element of $\mathbb{F}_7$ we easily see that this is irreducible. What about separable? Can we look at the formal derivative $f’(x)=3x^2$ which has a double zero at $0$ and hence gcd$(f,f’)=1$ and so $f(x)$ is […]

How many automorphisms of $\Bbb Z \oplus \Bbb Z_2$

The question is how many isomorphisms (of groups) exist from $\Bbb Z \oplus \Bbb Z_2$ to itself. I tried to define isomorphism by taking generators to generators, but I want to see convincing proof for that. Thanks

Subgroups of $G$, $|G| = p^n$

If $G$ is a finite group, $|G| = p^n$ , $p$ is a prime, then how do I prove $G$ has a subgroup of order $p^k$, for each $k$, $1 \leq k \leq n$?

Groups of order 6

Show, without Lagrange theorem, that every group $G$ of order $6$ has an element of order $3$. I know that if $G$ has an element $a$ of order 6, then $a^2$ has order $3$, but I am not shure how to show that for the case where $G$ does not have an element of order […]

What does it mean when two groups commute?

This is probably an easy question, but I can’t find the definition in my book. Let $G$ be a group, and let $H$ and $N$ be subgroups. What does it mean for $H$ and $N$ to commute? I have two possibilities that come to mind: $(\forall h \in H)(\forall n \in N)[hn=nh]$ $HN=NH$, that is, […]

Dihedral subgroups of $S_4$

Prove that in $S_4$ there are $3$ groups that are isomorphic to $D_4$. I know that the $2$-sylows of $S_4$ should be subgroups of order $8$, but to prove it is a bit tricky for me Any help would be appreciated, thanks.