Assignment 5

Exercise $\PageIndex{1}$

1. If $\sigma \in A_n$ and $\tau \in S_n\text{,}$ show that $\tau^{-1} \sigma \tau \in A_n\text{.}$
2. What are the possible cycle structures of elements of $A_5\text{?}$ What about $A_6\text{?}$

Answer

1. Let $\sigma \in A_n$ and $\tau \in S_n\text{.}$

 If $\tau \in A_n$ then $\tau^{-1} \sigma \tau \in A_n\text{.}$ Otherwise  $\tau$ is odd permutation and  $\tau^{-1}$ is also odd (show). Then sum of odd numbers is even, $\tau^{-1} \sigma \tau \in A_n\text{.}$

2,  The possible cycle types in ${A_5}$ are:
 $11111 , 221, 311$
 The possible cycle types in ${A_6}$ are:
$111111 , 2211, 3111, 33, 42$

Exercise $\PageIndex{2}$

1. Find all of the subgroups in $A_4\text{.}$ What is the order of each subgroup?
2.  What are the conjugacy classes of $A_4$?
3. Let $H=\{ e, (12)(34), (13)(24), (14)(23)\}.$ Find the left cosets and right cosets of $H$ in $A_4.$

Answer

1.  Since $|A_4|=12$, by Lagrange's theorem, the possible order of subgroups are $1, 2, 3, 4, 6$ and $12$. But $A_4$ has no subgroup of order $6$ (done in class).
Hence, the subgroups of $A_4$  are:
 Order $1: \{e\}$
Order $2: \{e, (12)(34)\}, \{e, (13)(24)\}, \{e, (14)(23)\}$
Order $3: \{e, (123), (321)\}, \{e, (124), (421)\}, \{e, (134), (431)\}, \{e, (234), (431)\}$
 Order $4: \{e, (12)(34), (13)(24), (14)(23)\}$
 Order $12: A_4$

2. Show that the conjugacy classes of $A_4$ are:
 $\{e\}, \{(12)(34), (13)(24), (14)(23)\}, \{(123), (134), (243), (142)\}, \{(321), (431), (234), (124)\}.$

3.  Since $[A_4:H]=\dfrac{12}{4}=3$. Hence there are $3$ left cosets of $H$ in $A_4$ and $3$ right cosets of $H$ in $A_4$
Left cosets:
$eH = H = (12)(34)H = (13)(24)H = (14)(23)H$
 Since $(123)(12)(34) = (134)$, $(123)(13)(24) = (243)$, $(123)(14)(23) = (142),$
 $(123)H = (134)H = (243)H = (142)H = \{(123), (134), (243), (142)\}$
  Since $(132)(12)(34) = (234)$, $(132)(13)(24) = (124)$, $(132)(14)(23) = (143).$
 $(132)H = (234)H = (124)H = (143)H = \{(132), (234), (124), (143)\}$
 Right cosets:
 $He = H = H(12)(34) = H(13)(24) = (14)(23)H$
 Since $(12)(34)(123) = (243)$, $(13)(24)(123) = (142)$, $(14)(23)(123) = (134),$
 $H(123) = H(243) = H(142) = H(134) = \{(123), (243), (142), (134)\}$
 Since $(12)(34)(132) = (143)$, $(13)(24)(132) = (234)$, $(14)(23)(132) = (124),$
$(132)H = (143)H = (234)H = (124)H  = \{(132), (143), (234), (124)\}.$

Exercise $\PageIndex{3}$

1. Find the center of $D_4$ and $D_5$. Justify your answer.
2. What is the center of $D_n\text{?}$
3.  Find the conjugacy classes of  $D_4$. Justify your answer.

Answer

1. Consider $a = s^r k$ and $b = r^j$ in $D_n$ for some $0 \leq k, j < n$. Then
$$ab = s^{r(k+j)},$$
and,
$$ba = s^{r(k-j)}.$$
The equation $ab = ba$ implies that 
$$k - j \equiv k + j \pmod{n}.$$
But this is only true if $j = \frac{n}{2}$ or $j = 0$.
Therefore, all reflections are not in the center of $D_n$. Furthermore, the only rotation that is in the center is $r^{\frac{n}{2}}$ (and trivially $e$).
So, if $n$ is odd, then the center of $D_n$ is $\{ e \}$, otherwise, the center of $D_n$ is $\{ e, r^{\frac{n}{2}} \}$.
For example,
$$Z(D_4) = \{ e, r^2 \}, \quad Z(D_5) = \{ e \}.$$

2. The center of $D_n$ is $\{e\}$ if $n$ is odd and $\{e, r^{n/2}\}$ if $n$ is even.

3. 

Let $D_4 = <r, s | r^4 = e, s^2 = e, srs = r^{-1}>.$
The conjugacy classes of $D_4$ are $\{e\}, \{r^2\}, \{r,r^3\}, \{s, r^2s\}, \{rs, r^3s\}$.
Clearly, $[e] = \{e\}$.

Consider the following for the class $[r^m]$ for some $m$:
$$r^n r^m r^{-n} = r^m,$$
and
$$s r^n r^m r^{-n} s = r^{-m},$$
for all $n$. Therefore, $[r^m] = \{r^m, r^{-m}\}$. That is, 
$$[r] = \{r, r^3\}, \quad [r^2] = \{r^2\}.$$

Consider the class $[sr^m]$ for some $m$. We have:
$$r^n s r^m r^{-n} = s r^{m - 2n},$$
and
$$s r^n s r^m s r^{-n} = s s r^{-n + m + n} = s r^{-m}.$$
for all $n$. Since $D_4$ has an even number of rotations, we can partition the reflections into two conjugacy classes: even and odd. If there were only one conjugacy class of reflections, $m - 2n$ would not cycle properly. Thus, we have:
$$[s r^2] = \{ s r^n \mid n \in 2\mathbb{N} \} = \{s, sr^2\},$$
and
$$[s r] = \{ s r, s r^3 \}.$$

To summarize, the conjugacy classes are:
$$[e], [r], [r^2], [s], [sr].$$
Thus, the members of a conjugacy class of $D_4$ are different but have the same type of effect on a square: 

1.   $r$ and $r^3$ are a 90-degree rotation in some direction,
2.    $s$ and $r^2s$ are a reflection across a diagonal,
3.  $rs$ and $r^3s$ are a reflection across an edge bisector.
    

Exercise $\PageIndex{4}$

List all of the subgroups of $S_4\text{.}$ Find each of the following sets:

1. $\displaystyle \{ \sigma \in S_4 : \sigma(1) = 3 \}$
2. $\displaystyle \{ \sigma \in S_4 : \sigma(2) = 2 \}$
3.  $\{ \sigma \in S_4 : \sigma(1) = 3$ and $\sigma(2) = 2 \}\text{.}$
4. Are any of these sets subgroups of $S_4\text{?}$

Exercise $\PageIndex{5}$

Let $\sigma \in S_X\text{.}$ If $\sigma^n(x) = y$ for some $n \in \mathbb Z\text{,}$ we will say that $x \sim y\text{.}$ 

1. Show that $\sim$ is an equivalence relation on $X\text{.}$
2.  Define the orbit of $x \in X$ under $\sigma \in S_X$ to be the set
   $${\mathcal O}_{x, \sigma} = \{ y : x \sim y \}\text{.} \nonumber$$

   Compute the orbits of each element in $\{1, 2, 3, 4, 5\}$ under each of the following elements in $S_5\text{:}$

   $$\begin{align*} \alpha & = (1254)\\ \beta & = (123)(45)\\ \gamma & = (13)(25)\text{.} \end{align*}$$
    That is for example: $$\begin{align*} {\mathcal O}_{x, \sigma}&  = \{ y \in \{1,2,3,4,5\} : x \sim y \}\\  &=\{ y :\sigma^n(x) = y,\text{for some } n\in \mathbb{N} \}\\  \end{align*}$$

3. If ${\mathcal O}_{x, \sigma} \cap {\mathcal O}_{y, \sigma} \neq \emptyset\text{,}$ prove that ${\mathcal O}_{x, \sigma} = {\mathcal O}_{y, \sigma}\text{.}$ The orbits under a permutation $\sigma$ are the equivalence classes corresponding to the equivalence relation $\sim\text{.}$
   A subgroup $H$ of $S_X$ is transitive if for every $x, y \in X\text{,}$ there exists a $\sigma \in H$ such that $\sigma(x) = y\text{.}$ Prove that $\langle \sigma \rangle$ is transitive if and only if ${\mathcal O}_{x, \sigma} = X$ for some $x \in X\text{.}$