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Prove that the intersection of two subgroups of a group is again a subgroup

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goni7g | eNotes Newbie

Posted August 24, 2013 at 4:08 PM via web

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Prove that the intersection of two subgroups of a group is again a subgroup

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embizze | High School Teacher | (Level 1) Educator Emeritus

Posted August 24, 2013 at 4:50 PM (Answer #1)

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Prove that the intersection of two subgroups is a subgroup:

To show that a subset with an operation is a subgroup, we need to show that ` a,b in H => ab^(-1) in H` where H is the subset of a group.

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Let `H,K subset G` . Let `M=H cap K` .

Suppose `a,b in M` ; then `a in H,b in H, a in K, b in K` .

Since H is a subgroup, `a,b in H ==> ab^(-1) in H`

``Since K is a subgroup, `a,b in K ==> ab^(-1) in K`

`a,b in M ==> a,b in H; a,b, in K` thus `ab^(-1) in M` .

** We know that the intersection is nonempty, since H,K are subgroups implies that they both have the identity element of the group as their identity element. **

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