Abstract Algebra/Group Theory/Subgroup/Subgroup Inherits Identity
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Theorem[edit]
Let H be subgroup of Group G. Let be the binary operation of both H and G
 H and G shares identity
Proof[edit]

0. 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.