Articles of group theory

Computing Invariant Subspaces of Matrix Groups

Does anyone have a program written in Mathematica (or SAGE or GAP) that computes the invariant subspace lattice of a matrix group?

In which of the finite groups, the inverse of Lagrange's Theorem is not correct?

This is a multiple choice for finite groups. For which one of the following groups, the converse of Lagrange’s Theorem is not generally satisfied? I know the converse is true for cyclic groups. 1) All abelian groups 2) All groups of order 8 3) The group $S_4$ 4) All groups of order 12 Thank you […]

Showing that the diagonal of $G \times G$ is maximal, where $G$ is simple

I have been trying to prove the following: Let $G$ be simple, and write $\Gamma=G \times G$. Let $D \le \Gamma$ be the diagonal subgroup, which consists of all elements of the form $(x,x)$, where $x \in G$. Show that $D$ is a maximal subgroup of $\Gamma$. As a hint I am given: Write $\Gamma=A […]

What are the finite order elements of $\mathbb{Q}/\mathbb{Z}$?

I need to find what are the at the group $\mathbb{Q}/\mathbb{Z}$. I think that any element at this group has a finite order, but I don’t know how to prove it… I’d like to get help with the proof writing… If I’m wrong, I’d like to to know it too… BTW: $\mathbb{Z}=(\mathbb{Z},+)$ $\mathbb{Q}=(\mathbb{Q},+)$ Thank you!

Cayley's Theorem – Questions on Proof Blueprint

Not a duplicate of this exquisite answer, the numbers in which I abide by here. Not querying the proof, hence please don’t discourse on it. Proof blueprint: Steps 1-2 in words. Left multiplication of every element in the group by a fixed element in the group constitutes a permutation of those elements. Steps 1-2 in […]

Painting a cube with 3 colors (each used for 2 faces).

A cube is about to get fully painted using $3$ different colors. Each color is being used for $2$ faces of a cube. How many different cubes can be created this way? I saw this in a fifth grade math contest and it does not appear to be an easy problem. It took me almost […]

Is there a convenient way to show that the symmetric group $S_4$ has a subgroup of order $d$ for each $d|24$?

Show that the symmetric group $S_4$ has a subgroup of order $d$ for each $d|24$. From Lagrange’s theorem I know that if $G \le S_4$, then the order of $G$ necessarily divides $|S_4|=24$. However the question actually asks the converse of the Lagrange’s theorem, so I cannot apply the theorem directly. (And I don’t think […]

Image of a normal subgroup under a homomorphism

I want to find an example of a group homomorphism $f:G\to H$ such that $A$ is a normal subgroup of $G$, but $f(A)$ is not so in $H$. Definitely the groups must be noncommutative and the function should not be onto. Please help me.

Abelianization of free product is the direct sum of abelianizations

I define $\text{Ab}(G)=G/[G,G]$ where $[G,G]$ is the commutator subgroup. I want to show that $$\text{Ab}(G_1*G_2)\cong \text{Ab}(G_1)\oplus\text{Ab}(G_2)$$ This page gives a categorical proof, but I don’t know much category theory. Can someone give a purely group-theoretic proof of this (I know the universal property of abelianizations)? By the universal property, it would suffice to show that […]

prove: A finitely generated abelian group can not be isomorphic to a proper quotient group of itself.

Let $G$ be a finitely generated abelian group. Then prove that it is not isomorphic to $\frac{G}{N}$, for every subgroup $N\neq\langle 1\rangle$.