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. ${\color {OliveGreen}e_{H}}\ast {\color {OliveGreen}e_{H}}={\color {OliveGreen}e_{H}}$	eH is identity of H (usage 1, 3)
2. ${\color {OliveGreen}e_{H}}\in H$	eH is identity of H (usage 1)
3. $H\subseteq G$	H is subgroup of G
4. ${\color {OliveGreen}e_{H}}\in G$	2. and 3.
5. ${\color {OliveGreen}e_{H}}\ast {\color {Blue}e_{G}}={\color {OliveGreen}e_{H}}$	4. and eG is identity of G (usage 3)
6. ${\color {OliveGreen}e_{H}}\ast {\color {Blue}e_{G}}={\color {OliveGreen}e_{H}}\ast {\color {OliveGreen}e_{H}}$	1. and 5.
7. ${\color {Blue}e_{G}}={\color {OliveGreen}e_{H}}$	cancellation on group G

Usages

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