# Theorem

Let H1, H2, ... Hn 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 $(H_{1}\cap H_{2})$ 