Groups and Group Actions: Lecture 5

In which we find that the alternating group is a group, and study subgroups in more detail.

Theorem 15: (i) Any permutation in $S_n$ can be written as a product of transpositions. (ii) A permutation cannot be both even and odd. We proved (i) by noting that since any permutation can be written as a product of disjoint cycles (Theorem 9), it suffices to show that any cycle can be written as a product of transpositions. And we have $(a_1 \, a_2 \, \dotsc \, a_k) = (a_1 \, a_2) (a_1 \, a_3) \dotsm (a_1 \, a_k)$ . For (ii), we showed that if $\sigma$ can be written as a product of $k$ transpositions, then $\det(P_{\sigma}) = (-1)^k$ .
Definition of the alternating group $A_n$ .
Proposition 16:
1. $A_n$ is a subgroup of $S_n$ .
2. For $n \geq 2$ , the order of $A_n$ is $\frac{1}{2} n!$ .
3. For $n \geq 4$ , $A_n$ is non-Abelian.
Proving this was fairly straightforward. To check the first part, we used our observation that $\sigma$ is even if and only if $\det(P_{\sigma})= 1$ . For the second part, we wrote down a bijection between $A_n$ and $S_n \setminus A_n$ . And for the third part, we noted that $(1 \, 2 \, 3)$ and $(1 \, 2 \, 4)$ are even permutations that don’t commute.
Proposition 17 (Subgroup test): Let $G$ be a group. The subset $H \subseteq G$ is a subgroup of $G$ if and only if $H$ is non-empty and $h_1 h_2^{-1} \in H$ for all $h_1$ , $h_2 \in H$ . We noted that if $H$ is a subgroup then since it contains the identity it is non-empty, and since it contains inverses and is closed under the group operation we must have $h_1 h_2^{-1} \in H$ whenever $h_1$ , $h_2 \in H$ . For the reverse direction, we said that if $h \in H$ then $h h^{-1} = e \in H$ so $H$ contains the identity, and then for any $h_1 \in H$ we have $e h_1^{-1} = h_1^{-1} \in H$ so $H$ contains inverses. Then for $h_1$ , $h_2 \in H$ , we have $h_2^{-1} \in H$ and so $h_1 (h_2^{-1})^{-1} = h_1 h_2 \in H$ , so $H$ is closed under the group operation.
Proposition 18: Let $G$ be a group. Let $H$ , $K$ be subgroups of $G$ . Then $H \cap K$ is a subgroup of $G$ . This is an exercise on Sheet 3.
Definition of the subgroup $\langle S \rangle$ generated by a subset $S$ of a group $G$ . Definition of the elements of $S$ as the generators of $\langle S \rangle$ .
Division algorithm: Let $a$ , $b$ be integers with $b > 0$ . Then there are unique integers $q$ and $r$ such that $a = qb + r$ and $0 \leq r < b$ .
Proposition 19: Let $G$ be a group. Take $g \in G$ .
1. We have $\langle g \rangle = \{ g^k : k \in \mathbb{Z}\}$ .
2. If $g$ has finite order, then $\langle g \rangle = \{e, g, g^2, \dotsc, g^{o(g)-1}\}$ .
In each part, it was clear that $\langle g \rangle$ contains the other set. For the first part, we used the subgroup test to show that $\{ g^k : k \in \mathbb{Z}\}$ is a subgroup of $G$ , and then since $\langle g \rangle$ is the smallest subgroup containing $g$ , we get equality. For the second part, we used the division algorithm to show that any $g^k$ (with $k \in \mathbb{Z}$ ) is in $\{e, g, g^2, \dotsc, g^{o(g)-1}\}$ .

Understanding today’s lecture

Theorem 15 doesn’t claim that the number of transpositions is well defined, just the parity of the number of transpositions. If I can write a permutation as a product of 6 transpositions, then I may well be able to write it as a product of 8 transpositions (in fact, I can: write it as those six transpositions followed by $(1 \, 2)(1 \, 2)$ ). But I’ll never be able to write it as a product of 7 transpositions, or 101.

Can you show that $(1 \, 2 \, 3)$ and $(1 \, 3 \, 2)$ are not conjugate in $A_4$ ? Can you list the elements of $A_5$ ? (Well, don’t list them, there are lots, but list the cycle types and say how many elements there are with each cycle type.)

Can you prove Proposition 18 (the intersection of two subgroups is a subgroup)? Can you extend it to arbitrary intersections of subgroups (as mentioned in the remark following Proposition 18 in the lecture)?

In $S_3$ , what is the subgroup $\langle (1 \, 2 \, 3)\rangle$ ? What is the subgroup $\langle (1 \, 2), (1 \, 2 \, 3)\rangle$ ?

Preparation for Lecture 6

If $G$ is a cyclic group and $H$ is a subgroup of $G$ , must $H$ itself be cyclic?

We saw that $C_2 \times C_2$ is not cyclic. Is $C_m \times C_n$ ever cyclic?

We’re going to need to talk about equivalence relations soon, so this would be a good time to remind yourself what an equivalence relation is. Do you have some favourite examples of equivalence relations?

One Response to “Groups and Group Actions: Lecture 5”

Groups and Group Actions: Lecture 10 | Theorem of the week Says:
April 26, 2017 at 11:48 am
[…] We proved this using the subgroup test. […]

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