Articles of group theory

Possible orders of normal subgroups using only the elements in $G$ and their orders

I am told some information about a group $G$ of order $168$. All we are told about G is that: It has one element of order one, $21$ elements of order $2$, $56$ elements of order $3$, $42$ elements of order $4$ and $48$ elements of order $7$. and later it will be proved to […]

Sorting numbers in a matrix by moving an empty entry through other entries is not always possible .

let $\mathbf{A}$ be the set of all $n\times n$ matrices on $\mathbb{N}_{n^2}=\{1,2,…,n^2\}$ with distinct entries. let $T$ be the set of all permutations of $\mathbf{A}$ which swap the entry 1 with one of its adjacent entries. adjacent means one is above/below/right/left of the other (and both are neighbors). Let $S$ be the permutation subgroup generated […]

$SL(3,\mathbb{C})$ acting on Complex Polynomials of $3$ variables of degree $2$

So I’m given the following definition: $h(g)p(z)=p(g^{-1}z)$ where g is an element of $SL(3,\mathbb{C})$, $p$ is in the vector space of homogenous complex polynomials of $3$ variables and $z$ is in $\mathbb{C}^3$. What I’m having trouble showing is that mapping $g$ to $h(g)$ is a group homomorphism. Namely, I know that $h(ab)p(z)=p(b^{-1}a^{-1}z)$, but I can’t […]

Prove that Q has an automorphism of order 3.

Let $$A=\begin{pmatrix}i & 0\\ 0 & -i\end{pmatrix}$$ and $$B=\begin{pmatrix}0 & 1\\ -1 & 0\end{pmatrix}.$$ Let $Q=\langle A,B\rangle.$ Prove that Q has an automorphism of order 3.

If $H$ is a subgroup of $\mathbb Q$ then $\mathbb Q/H$ is infinite

I’m trying to work out this question: Prove that if $H$ is a proper subgroup of $\mathbb{Q}$ then $\mathbb{Q}/H$ is infinite, but each of its elements have finite order. I thought, for the first part, that I could assume for contradiction that $\mathbb{Q}/H$ is finite of order $n$, then for all $\dfrac{a}{b}\in\mathbb{Q}$, $\dfrac{a^n}{b^n}$ is in […]

$p$-group and normalizer

Here is the question: a) Show that if $p$ is a prime number and $P$ is a $p$-subgroup of a finite group $G$, then $[G:P]=[N_G(P):P]$(mod p), where $N_G(P)$ denotes the normalizer of $P$ in $G$. b) Assume that $H$ is a subgroup of a finite group $G$, and that $P$ is a Sylow $p$-group of […]

Uniqueness of subgroups of a given order in a cyclic group

I am currently studying Serge Lang’s book “Algebra”, on page 25 it is proved that if $G$ is a cyclic group of order $n$, and if $d$ is a divisor of $n$, then there exists a unique subgroup $H$ of $G$ of order $d$. I have trouble seeing why the proof (as explained below) settles […]

Using GAP to compute the abelianization of a subgroup

Let $K$ be the group generated by four elements $x_1,\cdots,x_4$ with relations that each generator commutes with all its conjugates. (An equivalent relation is, any simple commutator with repeated generator is trivial; for example, $[[x_2,[x_1,x_3]],x_3]=1$.) It can be proved that $K$ is finitely presented. Let $A$ be the subgroup of $K$ generated by the following […]

On the number of Sylow subgroups in Symmetric Group

If $G$ is a finite group, and $P$ is a Sylow-$p$ subgroup of $G$, then the number of Sylow-$p$ subgroups in $G$ is at most $|G|/|P|$. In the Symmetric group $S_n$, the bound is attained only for certain Sylow subgroups. In $S_4$, the number of Sylow-$2$ subgroups is $|S_4|/2^3$. In $S_3$, the number of Sylow-$3$ […]

Calculating the order of an element in group theory

Calculate the order of the elements a.) $(4,9)$ in $\mathbb{Z_{18}} \times \mathbb{Z_{18}}$ b.) $(8,6,4)$ in $\mathbb{Z_{18}} \times \mathbb{Z_{9}} \times \mathbb{Z_{8}} $ I have the solutions given as a.) $(4,9)^{18}= (0,0)$ so the order is 18 OR the orders of 4,9, in their respective groups are 9 and 2, so the lowest common multiple is 18 […]