2
votes
1answer
26 views

Group divisibility question

I have the following question which I can't make sense of, here is the entire question: If $G$ is a group, $b\in G, o(b)=k$ and $b^n = e$, show that $k|n$ What is $o(b)$? Please help.
1
vote
4answers
57 views

Help with groups

let $G$ be a finite group with $e$ Identity element and let $a$ and $b$ belong to $g$ prove that if: $\gcd(o(a),o(b)) =1$ then $\langle a \rangle \cap \langle b \rangle = \{ e \}$. if someone can ...
1
vote
2answers
72 views

Why define $\gcd(r,s)$ as a positive generator $d$ of the cyclic group $H=\{nr+ms|n,m\in\mathbb{Z}\}$?

This is in regards to definition 6.8, p. 62 from Fraleigh's "A first course in abstract algebra". 6.8 Definition Let $r$ and $s$ be two positive integers. The positive generator $d$ of the ...
2
votes
1answer
32 views

Solving a divisibility problem using group theory

Inspired by this I decided to show this: Let $P_n=\displaystyle\prod_{1\leq i<j\leq n} \big(x_i-x_j\big)$ where $x_1\,\dots\, x_n$ are arbitrary integers. Prove $n!\,\big|\, 2P_n$. I am ...
1
vote
2answers
44 views

Help with a proof envolving a finite group and a specific bijection

Let $G$ be a finite group, and let $k>1$ be an integer. I need to prove that if the mapping $f:G\rightarrow G$, defined by $f(g)=g^k$, is bijection, then $\gcd(k,|G|)=1$. I almost certain that if ...
3
votes
0answers
76 views

Find the fundamental group and the Alexander polynomial

I would like to find the Alexander polynomial of the link $L$, described below. Let $K(q,r)$ be the $(q,r)$-torus knot embedded on a torus $V$. Inside the torus $V$, consider a smaller solid torus ...
3
votes
1answer
95 views

What can we say about the order of a group given the order of two elements?

If I know that a group of finite order has two elements $a$ and $b$ s.t. their orders are $6$ and $10$, respectively. What statements can be made regarding the order of the group? I know by ...
5
votes
5answers
105 views

How to prove $x \in H$

How to prove that Let H be a normal subgroup of a finite group G. If $\gcd(|x|, |G/H|)$ = 1, show that $x \in H$.
0
votes
2answers
82 views

Normal subgroup and divisibility

I am very bad with problems involving divisibility of orders and such. If anyone can give me some help with the following problem, it will be very much appreciated: Prove that every subroup $H$ of ...
1
vote
2answers
98 views

Homomorphism $\mathbb{Z}^2 \to \mathbb{Z}^2$ and g.c.d.

Let $f: \mathbb{Z}^2 \to \mathbb{Z}^2$ be a group homomorphism and suppose that $f(a,b) = (c,d)$. Prove: $\gcd(a,b) \mid \gcd(c,d)$. Prove: $\gcd(a,b) = \gcd(c,d)$ if $f$ is an ...
3
votes
3answers
243 views

Which Digit-Permutations Preserve Divisibility?

This is a completely random question that just happened to come to mind recently and I was wondering if the MathSE community had anything to say about it. Let $n > 1,b > 1$ be integers and ...