411 REFERENCE · Wikibooks
Abstract Algebra/Group Theory/Subgroup/Subgroup Inherits Identity
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Theorem
Let H be subgroup of Group G.
Let be the binary operation of both H and G
H and G shares identity
Proof
{| Style="width:60%"
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|0. Let eH, eG be identities of H and G respectively.
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1.
|| eH is identity of H (usage 1, 3)
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2.
|| eH is identity of H (usage 1)
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3.
|| H is subgroup of G
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4.
||2. and 3.
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5.
||4. and eG is identity of G (usage 3)
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6.
||1. and 5.
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7.
|| cancellation on group G
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Usages
Usage1 If H is subgroup of group G, identity of G is identity of H.
Usage2 If H is subgroup of group G, identity of G is in H.