Articles of group theory

An automorphism of order 2 which fixes only the identity

Let $G$ be a group. Assume there is an element $\phi\in\text{Aut}(G)$ such that $\phi(x)=x\Rightarrow x=e$, where $e$ is the identity of $G$, and that $\phi^2$ is the identity automorphism on $G$. I need to show that $G$ is abelian. My attempt: Let $\sim$ be a relation defined on $G$ as follows: For $x,y\in G$, we […]

Show $\operatorname{Aut}(C_2 \times C_2)$ is isomorphic to $D_6$

Show $\operatorname{Aut}(C_2 \times C_2)$ is isomorphic to $D_6$ (the group with $x^3=1$, $y^2=1$ and $xy=yx^2$). I’m not really sure how to express the elements of $\operatorname{Aut}(C_2 \times C_2)$. Would it be sufficient to show the elements of $\operatorname{Aut}(C_2 \times C_2)$, find their order and show they bijectively map to every element of $D_6$ and satisfy […]

A finite $p$-group has normal subgroup of index $p^2$

Someone suggested me to show by induction something more general is valid, i.e. if $|G|=p^n$ then $\forall\; 0\le i\le n\;\;\exists\; H_i\unlhd G$ s.t. $|H_i|=p^i$. Lookin’ at we can show that does exists subgroups of order $p^i\;\;\forall 0\le i\le n$, but the normality fails an half! I mean that I can prove the normality only […]

The conjugacy problem of finitely generated free group

I would like references for algorithms solving the conjugacy problem in $F_n$ (the free group on $n$ generators)?

Isomorphism between $SL(2,\mathbb{Z}) \times \mathbb{Z_2}$ and $GL(2,\mathbb{Z})$

Since $SL(2,\mathbb{Z})=\{A\in M_{(2,2)}(\mathbb{Z})|\det(A)=1\}$ and $GL(2,\mathbb{Z})=\{A\in M_{(2,2)}(\mathbb{Z})|\det(A)=\pm1\}$, one can naturally guess there may exist an isomorphism between $SL(2,\mathbb{Z}) \times \mathbb{Z_2}$ and $GL(2,\mathbb{Z})$. In my text book, the author shows that $SL(2,\mathbb{Z})\cong \mathbb{Z_4}*_{\mathbb{Z_2}}\mathbb{Z_6} $ and $GL(2,\mathbb{Z})\cong (\mathbb{Z_4}*_{\mathbb{Z_2}}\mathbb{Z_6})\times \mathbb{Z_2}$. Hence there should be such an isomorphism. But I failed to construct a conscise isomorphism directly from $SL(2,\mathbb{Z}) \times […]

Showing the existence of an eigenvector using groups

Let $\text{GL}_2(\mathbb{F}_p)$ act on $\mathbb{F}_p^2$ ($2$-vectors with entries in $\mathbb{F}_p$) by matrix multiplication: $$\mathbf{M}\in\text{GL}_2(\mathbb{F}_p),\;\mathbf{v}\in\mathbb{F}_p^2:\quad \mathbf{M}(\mathbf{v})=\mathbf{Mv}$$ I would like to prove the following claim: If $\mathbf{A}\in\text{GL}_2(\mathbb{F}_p)$ has $\text{ord}\,\mathbf{A}=p$ then there exists non-zero $\mathbf{w}\in\mathbb{F}_p^2$ such that $\mathbf{A}(\mathbf{w})=\mathbf{w}$ The hint given is to use the orbit-stabiliser theorem. I think the idea is to show that for some $\mathbf{w},\;p$ […]

Unique minimal subgroup

We are in the contest of the classification of all groups of order $2^3$. We know that a group $G$ with a unique maximal subgroup is necessarely cyclic. 1)Then my teacher said that the dual proposition is “almost” true: a group $G$ with a unique minimal subgroup is a $p$-group and if $p\neq2$ it’s also […]

Proving that the order of $(a,b)$ is the lcm of $|a|,|b|$

I want to solve the following exercise from Dummit & Foote’s Abstract Algebra text: Prove that the elements $(a,1)$ and $(1,b)$ of $A \times B$ commute and deduce that the order of $(a,b)$ is the least common multiple of $|a|$ and $|b|$. The first part is easy: $$(a,1)(1,b)=(a \star 1,1 \diamond b)=(a,b)=(1 \star a,b \diamond […]

There exist Sylow subgroups $P$ and $Q$ for which $= = p$.

From D&F’s sylow theory section: Show that if $n_p\not\equiv 1 \mod p^2$ then there are distinct Sylow $p$-subgroups $P$ and $Q$ of $G$ for which $[P:P \cap Q]=[Q : P \cap Q] = p$. Are automorphism groups required for this question? Does it involve permutation represntations or does it follow directly from the Sylow theorems?

Prove this set is a group.

This question already has an answer here: {5,15,25,35} is a group under multiplication mod 40 5 answers