411 REFERENCE · Wikibooks
Abstract Algebra/Group Theory/Subgroup/Intersection of Subgroups is a Subgroup
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Theorem
Let H1, H2, ... Hn be subgroups of Group G with operation
with is a subgroup of Group G
Proof
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| 1.
|| H1 is subgroup of G
|-
| 2.
|| H2 is subgroup of G
|-
| 3.
|| 1. and 2.
|}
with is a Group
Closure
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| 4. Choose
||
|-
| 5.
|| closure of H1
|-
| 6.
|| closure of H2
|-
| 7.
|| 5. and 6.
|}
Associativity
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| 8. is associative on G.
|| Group G's operation is
|-
| 9.
|| 3.
|-
| 10. is associative on
|| 8. and 9.
|}
Identity
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| 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
|}
Inverse
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| 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
|}