Let $G$ a finite group and $H$ subgroup of index $2$. Let $x\in H$ so that the number of conjugates of $x$ in $G$ is $n$. Show that the the number of conjugates of $x$ in $H$ is $n$ or $n/2$.

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 […]

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 […]

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 […]

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.

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 […]

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 […]

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 […]

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 […]

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-$3$ […]

Intereting Posts

Picard group and cohomology
Intuitive understanding of the Reidemeister-Schreier Theorem
Showing every knot has a regular projection using differential topology
How to count derangements?
Proving that every set $A \subset \Bbb N$ of size $n$ contains a subset $B \subset A$ with $n | \sum_{b \in B} b$
Turning a Piecewise Function into a Single Continuous Expression
Zero's of an ODE.
Is the derivative of an integral always continuous?
How to calculate generalized Puiseux series?
Measurable cardinal existence condition
Problem from Armstrong's book, “Groups and Symmetry”
Checking understanding on proving uniqueness of identity and inverse elements of a group.
Counting men and women around a circular table such that no 2 men are seated next to each other
How does Knuth's algorithm for calculating logarithm work?
Good introductory books on homological algebra