Abstract Algebra/Group Theory/Subgroup/Subgroup Inherits Identity

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Theorem[edit]

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


H and G shares identity

Proof[edit]

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[edit]

  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.