Articles of group theory

Prove two pairs of subspaces are in the same orbit using dimension

Let $V$ be a finite-dimensional vector space over a field $K$. Consider the group $GL(V)$ of non-singular linear maps acting on pairs of subspaces $(U,W)$ of fixed dimensions $p$, $q$ respectively by $g(U,W)=(gU,gW)$. Prove that two pairs $(U,W)$ and $(U’,W’)$ are in the same $GL(V)$-orbit (i.e., there exists $g\in GL(V)$ such that $gU=U’$ and $gW=W’$) […]

Check in detail that $\langle x,y \mid xyx^{-1}y^{-1} \rangle$ is a presentation for $\mathbb{Z} \times \mathbb{Z}$.

“Check in detail that $\langle x,y \mid xyx^{-1}y^{-1} \rangle$ is a presentation for $\mathbb{Z} \times \mathbb{Z}$.” Exercise 27.5 from “Groups and Symmetry” M.A.Armstrong. This should be an easy exercise but I’m completely unable to answer it. I know the group would be equal to the quotient group $F(X)/N$ where $X$ is an alphabet and $N$ […]

Is the question phrased properly? and is my proof correct? (An infinite alternating group is simple)

I’m interested in the following exercise from Dummut & Foote’s Abstract algebra text (p. 151) Let $D$ be the subgroup of $S_\Omega$ consisting of permutations which move only a finite number of elements of $\Omega$ (described in Exercise 17 in Section 3) and let $A$ be the set of all elements $\sigma \in D$ such […]

Characteristic subgroups of a direct product of groups

Let $G=H\times K$ and $H\times 1$ be a characteristic subgroup of $G$. Then can we conclude that $1\times K$ is also a characteristic subgroup of $G$? My motivation is the case where orders of $H$ and $K$ are relatively prime. In that case, both must be characteristic subgroups of $G$. So I wonder: if one […]

Number of transitive acting groups on four letters?

Constructing a group (a permutation group) which acts on a set of 4 letters transitively is easy. for example $G_{1}=\{id, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)\}$ < $S_{4}$. Can it be verified how many groups we have like $G_{1}$ ?

Group of order $48$ must have a normal subgroup of order $8$ or $16$

Prove a group of order $48$ must have a normal subgroup of order $8$ or $16$. Solution: The number of Sylow $2$-subgroups is $1$ or $3$. In the first case, there is a normal subgroup of order $16$ so we are done. In the second case, let $G$ act by conjugation on the Sylow $2$-subgroups. […]

Finding $A,B\in SL_2(\Bbb{Z})$ of finite order with the property that $AB=C$ where the order of $C$ is infinite.

I’m trying to think of two matrices $A,B\in SL_2(\Bbb{Z})$ of finite order ($A^n=B^m=I$) with the property that $AB=C$ where the order of $C$ is infinite. I guess that just by trial and error I could find two of those matrices, but I would like to find those in a little bit more sophisticated way. What […]

Is ${\mathbb Z} \times {\mathbb Z}$ cyclic?

Not sure where to go with this, but I don’t think it is cyclic..

Galois group of the extension $E:= \mathbb{Q}(i, \sqrt{2}, \sqrt{3}, \sqrt{2})$

In order to make a smaller example for my question Galois group of the field of all constructible complex numbers, I am posing this new question. I know already, that E is a galois extension of $\mathbb{Q}$ of dimension 16, it being a galois extension of the splitting field $L$ of the polynomial $X^4-2$ of […]

Groups satisfying the normalizer condition

I have two questions related to nilpotent groups: Is the class of groups satisfying the normalizer condition closed under taking quotients? Are there examples of (infinite) groups satisfying the normalizer condition but not solvable? Thanks. EDIT: Here ‘the normalizer condition’ is the condition that the normalizer of any proper subgroup properly contains it.