## Groups and Group Actions: Lecture 5

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

• 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 the fact from last time 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 $hh^{-1} = e \in H$ so $H$ contains the identity, and then for any $h_1 \in H$ we have $eh_1^{-1} = h_1^{-1} \in H$ 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 $\{e, g, g^2, \dotsc, g^{o(g)-1}\}$.

• Theorem 20: Let $G$ be a cyclic group.
1. If $G$ is finite, with $|G| = n$, then $G$ is isomorphic to $C_n$.
2. If $G$ is infinite, then $G$ is isomorphic to $\mathbb{Z}$.

The first part was clear from our previous work. For the second part, we defined a map $\theta : G \to \mathbb{Z}$ that sends $g^k$ to $k$.  Checking that this is an isomorphism is an exercise.

### Understanding today’s lecture

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 $\langle (1 \, 2), (1\, 2 \, 3) \rangle$?

You might like to find a book and look up the classification of conjugacy classes in the alternating group: it turns out to be rather interesting.

### 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?