Subgroups Summary

Subgroups Overview

• Definition of group: $(G, <em>)$ where $G$ is a set and $</em>$ is a binary operation.

• Subgroup condition: If $(H, *)$ is a group and $H \leq G$, then $H$ is a subgroup.

Examples of Subgroups

• The group of integers $\mathbb{Z}$ is a group under addition.

• The set of even integers $2\mathbb{Z}$ is a subgroup of $\mathbb{Z}$ under addition.

• The set of upper triangular $2 \times 2$ matrices with integer entries forms a subgroup under matrix addition.

• Unit circle $U^1$ is a group under multiplication.

Properties of Subgroups

• A subgroup must contain the identity element of the parent group.

• A subgroup must be closed under the group operation.

• Every element must have an inverse within the subgroup.

Subgroup Criterion

• For a subset $H$ of group $G$ to be a subgroup:1. Must contain identity element. 2. Must be closed under the group operation. 3. Must have inverses.

Types of Subgroups

• Improper subgroup: the whole group $G$.

• Nontrivial subgroups: contain elements other than just the identity.

• Cyclic subgroups: generated by a single element $a$, denoted $\langle a \rangle$.

Illustrative Examples

• Group of symmetries $D_n$ has subgroups formed by its rotational symmetries.

• Klein group is a subgroup of order 4 under addition.

Theorems

• Every cyclic group is a subgroup of itself and contains all integer powers of its generating element.

• Finite nonempty subsets of groups can be checked for the subgroup criteria effectively.

Closure Under Operations

• For groups $H$ and $G$, if $ab \in H$ for all $a,b \in H$, then $H$ is closed under the operation.

• The inverse property implies if $a \in H$, then $a^{-1} \in H$.

Conclusion

• Understanding subgroups provides insight into the structure of groups and their properties. Each subgroup has distinct characteristics that contribute to the overall behavior of the parent group.