Abstract Algebra/Group Theory/Subgroup/Subgroup Inherits Identity

Theorem

Let H be subgroup of Group G. Let $\ast$ be the binary operation of both H and G

H and G shares identity

Proof

0. Let eH, eG be identities of H and G respectively.
1. $e_H\ast e_H=e_H$	[[../../Group/Definition of a Group/Definition of Identity#Usage1|eH is identity of H (usage 1, 3)]]
2. $e_H\in H$	[[../../Group/Definition of a Group/Definition of Identity#Usage1|eH is identity of H (usage 1)]]
3. $H\subseteq G$	[[../Definition of a Subgroup#Usage2|H is subgroup of G]]
4. $e_H\in G$	2. and 3.
5. $e_H\ast e_G=e_H$	4. and [[../../Group/Definition of a Group/Definition of Identity#Usage3|eG is identity of G (usage 3)]]
6. $e_H\ast e_G=e_H\ast e_H$	1. and 5.
7. $e_G=e_H$	[[../../Group/Cancellation#Usage1|cancellation on group G]]

Usages

1. Template:Anchor If H is subgroup of group G, identity of G is identity of H.
2. Template:Anchor If H is subgroup of group G, identity of G is in H.