Abstract Algebra/Group Theory/Subgroup/Intersection of Subgroups is a Subgroup
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Contents
Theorem[edit]
Let H_{1}, H_{2}, ... H_{n} be subgroups of Group G with operation
 with is a subgroup of Group G
Proof[edit]
[edit]
with is a Group [edit]
Closure[edit]
Associativity[edit]


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

Identity[edit]


11. and Subgroup H_{1} and H_{2} inherit identity from G 12. e_{G} is identity of G, 13. and 9. 14. has identity e_{G} definition of identity

Inverse[edit]


15. Choose 16. , , and 17. g_{H1}^{1} in H1, and g_{H2}^{1} in H2. G, H_{1}, and H_{2} are groups 18. 19. G and H1 shares identity e 20. g_{H1}^{1} is inverse of g in G 19. and definition of inverse 21. Let g_{G}^{1} be inverse of g has in G 22. g_{G}^{1} = g_{H1}^{1} inverse is unique 22. g_{G}^{1} = g_{H2}^{1} similar to 21. 23. 24. g has inverse g^{1} in
