Identity of Subgroup

Theorem

Let $G$ be a group whose identity is $e$.

Let $H$ be a subgroup of group $G$.

Then the identity of $H$ is also $e$.

Proof

A group is a fortiori a monoid.

From the Cancellation Laws, all of its elements are cancellable.

The result then follows from Identity of Cancellable Monoid is Identity of Submonoid.

$\blacksquare$

Also see

• Identity of Submonoid is not necessarily Identity of Monoid

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 5.2$. Subgroups
• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Subgroups
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 36.2 \ \text{(i)}$: Subgroups
• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $4$: Subgroups: Proposition $4.2$: Remark $2$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): subgroup
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): subgroup