Question
Asked Sep 26, 2019
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Let G be a group, and let X be a set. Let I be the intersection of all subgroups of G that
contain X. Show that I is the smallest subgroup of G that contains X. Conclude that
1= (x)
I
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Let G be a group, and let X be a set. Let I be the intersection of all subgroups of G that contain X. Show that I is the smallest subgroup of G that contains X. Conclude that 1= (x) I

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Expert Answer

Step 1

Let G be a group and X be a set in G. Suppose I is the intersection of all subgroups of G that contains X.

To prove:

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I is the smallest subgroup ofG that contains X

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Step 2

Proof:

...
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Let I jN be all the subgroups ofG. Then I = NI,. J jeN As the intersection of subgroups ofa group is also a subgroup, then I = nI, is a jeN subgroup of G. As XCI nI This implies X is contained in each subgroup I, jeN. jeN As the intersection ofsets is the smallest of all sets considered, then I becomes the smallest subgroup to have Xcontained in it

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