Abstract Algebra/Group Theory/Subgroup/Subgroup Inherits Identity

TheoremEdit

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

H and G shares identity

ProofEdit

0. Let eH, eG be identities of H and G respectively.
1.  
eH is identity of H (usage 1, 3)
2.  
eH is identity of H (usage 1)
3.  
H is subgroup of G
4.  
2. and 3.
5.  
4. and eG is identity of G (usage 3)
6.  
1. and 5.
7.  
cancellation on group G

UsagesEdit

  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.