Definition:Subgroup

Definition

Let $\left({G, \circ}\right)$ be an algebraic structure.

Then $\left({H, \circ}\right)$ is a subgroup of $\left({G, \circ}\right)$ iff:

• $\left({H, \circ}\right)$ is a group
• $H$ is a subset of $G$.

This is represented symbolically as $H \le G$.

Note that in order for $\left({H, \circ}\right)$ to be a subgroup of $\left({G, \circ}\right)$, the operation on $G$ and $H$ must also be the same.

In the case of $\left({G, \circ}\right)$ and $\left({H, \circ}\right)$, the operation is $\circ$.

It is usual that $\left({G, \circ}\right)$ is itself a group, but that is not necessary for the definition.

If it is known that $\left({G, \circ}\right)$ is in fact a group, then one may verify if a subset is a subgroup by using either the one-step or two-step subgroup test, as well as by checking for each individual group property.

Sources

• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 5.2$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 8$
• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 1.9$
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): Chapter $\text{II}$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 35$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 36$
• John F. Humphreys: A Course in Group Theory (1996): $\S 4$: Definition $4.1$