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

< Abstract Algebra | Group Theory | Subgroup

# TheoremEdit

Let H_{1}, H_{2}, ... H_{n} be subgroups of Group G with operation

- with is a subgroup of Group G

# ProofEdit

# Edit

# with is a Group Edit

## ClosureEdit

## AssociativityEdit

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

## IdentityEdit

11. and Subgroup H _{1}and H_{2}inherit identity from G12. *e*_{G}is identity of G,13. and 9. 14. has identity *e*_{G}definition of identity

## InverseEdit

15. Choose 16. , , and 17. *g*_{H1}^{−1}in H1, and*g*_{H2}^{−1}in H2.G, H _{1}, and H_{2}are groups18. 19. G and H1 shares identity e 20. *g*_{H1}^{−1}is inverse of*g*in G19. and definition of inverse 21. Let *g*_{G}^{−1}be inverse of*g*has in G22. *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