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## Homework Statement

Prove that the intersection of any collection of subgroups of a group is again a subgroup

## Homework Equations

## The Attempt at a Solution

Fixed proof

Let [itex]H_1 and H_2 [/itex] be subgroups on G. We first see if [itex]H_1 \cap H_2[/itex] is again a subgroup. We see if [itex]a,b\in H_1 \cap H_2[/itex] then [itex] ab\in H_1 \cap H_2[/itex]. Thus [itex]H_1 \cap H_2[/itex] is closed. Automatically the identity element has to be in [itex]H_1 \cap H_2[/itex] since [itex]H_1 and H_2 [/itex] are subgroups. And if [itex]a\in H_1 \cap H_2[/itex] then it follows that [itex]a^{-1}\in H_1 \cap H_2[/itex]. Thus [itex]H_1 and H_2 [/itex] is a subgroup.

I know this argument may sound redundant and in my inductive step I noticed that I never really used my assumption but would this work as a proof?

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