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

Theorem

Let H₁, H₂, ... H_n be subgroups of Group G with operation $\ast$

$H_{1}\cap H_{2}\cap \cdots \cap H_{n}$ with $\ast$ is a subgroup of Group G

$\color {RawSienna}(H_{1}\cap H_{2})\subseteq G$

1. $H_{1}\subseteq G$	H1 is subgroup of G
2. $H_{2}\subseteq G$	H2 is subgroup of G
3. $(H_{1}\cap H_{2})\subseteq G$	1. and 2.

$\color {RawSienna}H_{1}\cap H_{2}$ with $\color {RawSienna}\ast$ is a Group

Closure

4. Choose $x,y\in (H_{1}\cap H_{2})$
5. $x\ast y\in H_{1}$	closure of H1
6. $x\ast y\in H_{2}$	closure of H2
7. $x\ast y\in (H_{1}\cap H_{2})$	5. and 6.

Associativity

8. $\ast$ is associative on G.	Group G's operation is $\ast$
9. $(H_{1}\cap H_{2})\subseteq G$	3.
10. $\ast$ is associative on $(H_{1}\cap H_{2})$	8. and 9.

Identity

11. $e_{G}\in H_{1}$ and $e_{G}\in H_{2}$	Subgroup H1 and H2 inherit identity from G
12. $\forall g\in G:e_{G}\ast g=g\ast e_{G}=g$	eG is identity of G,
13. $\forall \;g\in (H_{1}\cap H_{2}):e_{G}\ast g=g\ast e_{G}=g$	$(H_{1}\cap H_{2})\subseteq G$ and 9.
14. $(H_{1}\cap H_{2})$ has identity eG	definition of identity

Inverse

15. Choose $g\in (H_{1}\cap H_{2})\subseteq G$
16. $g\in H_{1}$, $g\in H_{2}$, and $g\in G$
17. gH1−1 in H1, and gH2−1 in H2.	G, H1, and H2 are groups
18. $g_{H1}^{-1}\in G$	$H_{1}\subseteq G$
19. $g_{H1}^{-1}\ast g=g\ast g_{H1}^{-1}=e_{G}$	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. $g^{-1}=g_{H1}^{-1}=g_{H2}^{-1}\in (H_{1}\cap H_{2})$
24. g has inverse g−1 in $(H_{1}\cap H_{2})$