Subsets and Subgroups

Definition

Let (G,\star) be a group.

(H,\star) is a subgroup of (G,\star) if

(1) (H,\star) is a group

(2) H is a subset of G.

This is notated H \leq G.

Definition

Let G be a group, and H be a subgroup of G. The subgroup H = G itself is the improper subgroup. All other subgroups are proper.

Definition

Let G be a group. The subgroup \{e_G\} is the trivial subgroup. All other subgroups are nontrivial.

Theorem

A subset H of a closed group G is a subgroup of G if and only if

(1) H is closed under the binary operation of G

(2) The identity e_G is in H

(3) For all a \in H, a^{-1} \in H.

Theorem

Let H be a finite nonempty subset

Cyclic Subgroups

Theorem

Let G be a group and let a \in G. Then

H = \{a^n \mid n \in \mathbb{Z}\}

is a subgroup of G and is the smallest subgroup of G that contains a. Furthermore, every subgroup containing a contains H.

Definition

Let G be a group and let a \in G. Then the subgroup above is known as the cyclic subgroup of G generated by a.

Definition

A group G is cyclic if and only if there exist some a such that \langle a\rangle = G.

Theorem

Every cyclic group is abelian

Proof

Let G be a cyclic group and let a generate G.

If g_1 and g_2 are any two elements of G, there exists integers r and s such that g_1 = a^r and g_2 = a^s. Then

g_1g_2 = a^ra^s = a^{r+s} = a^{s+r} = a^sa^r = g_2g_1.

So G is abelian.

Theorem (Division Algorithm for \mathbb{Z})

Let m be a positive integer and n be any integer. There exists integers q,r sich that

n = mq + r \quad\text{and}\quad 0 \leq r < m.

Theorem

A subset of a cyclic group is cyclic.

Proof

Let G be a cyclic group generated by a and let H be a subgroup of G.