Articles of group isomorphism

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 […]

Steps to prove or disprove if two rings are isomorphic

So i’m struggling on how to prove if two rings are not isomorphic to one another. My professor told me that if a ring is not isomorphic to another, the best way to prove that this is true is to find a preserved property of isomorphisms that is not held. So i considered the following: […]

The Group of order $p^3$

p is a prime number. Let G be the Group of order $p^3$ whose every element in $G$ has order p. Then G is not isomorphic to any subgroup of $GL(2; \mathbb{C})$. This claim is correcet? Please help me.

Does every automorphism of a permutation group preserve cycle structure?

In class we proved that the conjugacy map preserves cycle structure, and I was wondering if this was the case for any automorphism of a permutation group. I intuitively think that it should be the case, because automorphisms are isomorphisms from a group to itself and isomorphisms should not modify any fundamental structure of an […]

Proving that $D_{12}\cong S_3 \times C_2$

Prove that $D_{12}\cong S_3 \times C_2$. I really dont know how I should start this question. My gut feeling says in some way I have to consider normal subgroups of $D_{12}$ but I cannot see how this will lead necessarily to a unique solutions. No full solutions please hints only (partly because I cannot give […]

Showing that the groups (Q,+) and (Q⁺,*) are not isomorphic

This question already has an answer here: Group of positive rationals under multiplication not isomorphic to group of rationals 2 answers

Why is this group of matrices isomorphic to the dihedral group?

I am reading Abstract Algebra, Theory and Applications by Judson and in exercise $13$ chapter $9$, Isomorphisms, I need to prove that the set of matrices $$A=\pmatrix{ \omega & 0 \\ 0 & \omega ^{-1}} \qquad B=\pmatrix{ 0 & 1 \\ 1 & 0}$$ Where $\omega = e^{2\pi i /n}$ form a group isomorphic to […]

Intuition about the second isomorphism theorem

In group theory we have the second isomorphism theorem which can be stated as follows: Let $G$ be a group and let $S$ be a subgroup of $G$ and $N$ a normal subgroup of $G$, then: The product $SN$ is a subgroup of $G$. The intersection $S\cap N$ is a normal subgroup of $G$. The […]

Let $R$ be a commutative unital ring. Is it true that the group of units of $R$ is not isomorphic with the additive group of $R$?

Let $R$ be a commutative ring with unity, and let $R^{\times}$ be the group of units of $R$. Then is it true that $(R,+)$ and $(R^{\times},\ \cdot)$ are not isomorphic as groups ? I know that the statement is true in general for fields. And it is trivially true for any finite ring (as $|R^{\times}| […]

How to geometrically show that there are $4$ $S_3$ subgroups in $S_4$?

As shown in this note, the symmetry group $S_4$ for a cube has 4 subgroups that are isomorphic to $S_3$ for an equilateral triangle. How to geometrically illustrate this fact? Specifically, where are the equilateral triangles embedded in the cube? Related post: How to geometrically show that there are $3$ $D_4$ subgroups in $S_4$?