Articles of group isomorphism

Question on Smith normal form and isomorphism

Put $A=\begin{pmatrix} 1 & -5 & 4\\ 1 & -2 & 13\\ -2 & 13 & 7 \end{pmatrix}.$ The smith normal form of this matrix is \begin{pmatrix} 1 & 0 & 0\\ 0 & 3 & 0\\ 0 & 0 & 6 \end{pmatrix} and now I want to find $a , b, c$ $\in$ $\Bbb […]

A mapping $\phi:\mathfrak{so}(3) \to \mathbb{R}^{3}$ such that $\phi(gXg^{-1}) = g \phi(X)$

An element $X \in \mathfrak{so}(3)$ can be written generally written as $X = \left[ \begin{matrix} 0 & -z & y \\ z & 0 & -x \\ -y & x & 0 \end{matrix} \right]$. I’ve been given a mapping $\phi:\mathfrak{so}(3) \to \mathbb{R}^{3}$ which is defined by its components as: $$ \phi\left( \left[ \begin{matrix} 0 & […]

$G=\langle a,b \mid abab^{-1}\rangle$ and $H=\langle c,d \mid c^2d^2\rangle$ are isomorphic (Can't use Seifert/van Kampen Theorem)

I’m reading up on Algebraic Topology in preparation for a summer course, and learning about the classification of surfaces I ran across this problem: Show that the groups $G=\langle a,b \mid abab^{-1}\rangle$ and $H=\langle c,d \mid c^2d^2\rangle$ are isomorphic. It’s at a point in the text where Van Kampen’s theorem hasn’t yet been covered, so […]

If $G $ is a group of order $12$ not isomorphic to $A_4$ then does $G$ have an element of order $6$ ?

If $G $ is a group of order $12$ not isomorphic to $A_4$ then does $G$ have an element of order $6$ ? ( By Cauchy’s theorem I can show that there are elements of order $2$ and $3$ but cant proceed further , please help )

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.

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 […]

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 […]

Order of matrix and monic irreducible polynomial over finite field

I want to verify (and prove – in case it is true) the following proposition. Suppose $\mathbb{F}_p$ is a finite field and $m(x)$ is a monic irreducible polynomial over $\mathbb{F}_p$ with $\mathrm{deg}(m(x))=n$. If $\mathbf{A}$ is an $n \times n$ matrix over $\mathbb{F}_p$ whose characteristic polynomial is $m(x)$, then $\mathrm{ord}(\mathbf{A})=p^n-1$, where $\mathrm{ord}(\mathbf{A})$ denotes the least positive […]

What automorphisms exist on the abelian group of positive rationals under multiplication?

Consider the abelian group $(\mathbb{Q}_{>0}, \times)$. What automorphisms exist for this group? I can only think of the trivial one and of $\phi(q) = \frac{1}{q}$. If we relax the problem to injective homomophisms from $(\mathbb{Q}_{>0}, \times)$ to itself, do we get additional results?

How far can we go with group isomorphisms?

The following is quoted from Wikipedia: From the standpoint of group theory, isomorphic groups have the same properties and need not be distinguished. But this statement is too general. What I’m wondering about is whether there is a limit to these properties for which isomorphic groups “need not be distinguished”. For instance, I have been […]