Abstract Algebra/Group Theory/Subgroup/Intersection of Subgroups is a Subgroup

TheoremEdit

Let H1, H2, ... Hn be subgroups of Group G with operation  

  with   is a subgroup of Group G

ProofEdit

Edit

1.   H1 is subgroup of G
2.   H2 is subgroup of G
3.   1. and 2.

with is a Group Edit

ClosureEdit

4. Choose  
5.   closure of H1
6.   closure of H2
7.   5. and 6.

AssociativityEdit

8.   is associative on G. Group G's operation is  
9.   3.
10.   is associative on   8. and 9.

IdentityEdit

11.   and   Subgroup H1 and H2 inherit identity from G
12.   eG is identity of G,
13.     and 9.
14.   has identity eG definition of identity

InverseEdit

15. Choose  
16.  ,  , and  
17. gH1−1 in H1, and gH2−1 in H2. G, H1, and H2 are groups
18.    
19.   G and H1 shares identity e
20. gH1−1 is inverse of g in G 19. and definition of inverse
21. Let gG−1 be inverse of g has in G
22. gG−1 = gH1−1 inverse is unique
22. gG−1 = gH2−1 similar to 21.
23.  
24. g has inverse g−1 in