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

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# 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.