Calculate $g^2, g^3, f^2$

Question

Everyone. I am not sure how to start this problem. If anyone can show me a step by step process with an explanation I would appreciate it.

If $S= \{x_1,x_2,x_3,x_4\}$, let $f,g\in S_4$ be defined by: $$f:x_1\to x_2, x_2\to x_3, x_3\to x_4, x_4\to x_1$$ and: $$g:x_1\to x_2, x_2\to x_1, x_3\to x_3, x_4\to x_4$$

Hint: If $f$ sends $x_1\to x_2$ and $x_2 \to x_3$ then $f^2$ sends $x_1$ to $x_3$. — Derek Allums, Sep 06 '14 at 18:11

score 2 · Answer 1 · answered Sep 06 '14 at 18:19

It's customary to use this notation

$$g=\begin{pmatrix}1&2&3&4\\2&1&3&4\end{pmatrix}=(12)\leftarrow\text{transposition}$$ so $$g^2=\operatorname{id}\quad;\quad g^3=g$$ and by the same method we have $$f=(1234)\leftarrow\text{cycle}$$ so $$f^2=(13)(24)$$

score 1 · Accepted Answer · answered Sep 06 '14 at 18:12

1

Given $f(x)$:

$f(x_1)=x_2$
$f(x_2)=x_3$
$f(x_3)=x_4$
$f(x_4)=x_1$

Compute $f^2(x)$ as follows:

$f^2(x_1)=f(f(x_1))=f(x_2)=x_3$
$f^2(x_2)=f(f(x_2))=f(x_3)=x_4$
$f^2(x_3)=f(f(x_3))=f(x_4)=x_1$
$f^2(x_4)=f(f(x_4))=f(x_1)=x_2$

You can use this example in order to compute $g^2(x)$ and $g^3(x)$...

answered Sep 06 '14 at 18:12

barak manos

43,109

this question is assumin that x1 x2 .. are separate elements and not subsets. Am I right @barak manos – DSinghvi Sep 06 '14 at 18:15
@DSinghvi: Subsets of what? – barak manos Sep 06 '14 at 18:16
means subsets of any thing let say real numbers. what I wanted to say is they are discrete elements and since we x1 -> x2 can mean if x1 contain elements that each elemet would have its different image in x2 – DSinghvi Sep 06 '14 at 18:18
@barakmanos thanks! Very helpful! I understand what I am doing know. – Steven Lynn Sep 06 '14 at 18:20
@StevenLynn: You're welcome :) – barak manos Sep 06 '14 at 18:21

score 1 · Answer 3 · answered Sep 06 '14 at 18:14

1

The multiplication on $S_{n}$ is defined as function composition thus $$ f^{2}(x)=f(f(x)),g^{3}(x)=g(g(g(x))) $$

etc'

Just evaluate the above terms at $x_{i}$ with $1\leq i\leq4$ to determine what permutations of $S_{4}$ they are

answered Sep 06 '14 at 18:14

Belgi

23,150

Calculate $g^2, g^3, f^2$

3 Answers3