# Abstract Algebra/Group Theory/Subgroup/Subgroup Inherits Identity

# Theorem

editLet H be subgroup of Group G. Let be the binary operation of both H and G

- H and G shares identity

# Proof

edit0. Let e _{H}, e_{G}be identities of H and G respectively.- 1.

e _{H}is identity of H (usage 1, 3)- 2.

e _{H}is identity of H (usage 1)- 3.

H is subgroup of G - 4.

2. and 3. - 5.

4. and e _{G}is identity of G (usage 3)- 6.

1. and 5. - 7.

cancellation on group G

# Usages

edit- If H is subgroup of group G, identity of G is identity of H.
- If H is subgroup of group G, identity of G is in H.