# Intersection of two subgroups of a group is again a subgroup

Last Updated : 05 Mar, 2021

Group
It is a set equipped with a binary operation that combines any two elements to form a third element in such a way that three conditions called group axioms are satisfied, namely associativity, identity, and invertibility.

Subgroup
If a non-void subset H of a group G is itself a group under the operation of G, we say H is a subgroup of G.

To Prove :
Prove that the intersection of two subgroups of a group G is again a subgroup of G.

Proof :
Let H and H2 be any two subgroups of G.
Then,

H1 âˆ©  H2  â‰   âˆ…

Since at least the identity element ‘e’ is common to both H1 and H2 .
In order to prove that H âˆ© H is a subgroup, it is sufficient to prove that

   a âˆˆ H1 âˆ© H2 ,  b âˆˆ H1 âˆ© H2
â‡¢ a b-1 âˆˆ H1 âˆ© H2

Now,

   a âˆˆ H1 âˆ© H2
â‡¢ a âˆˆ H1  and   a âˆˆ H2
â‡¢ b âˆˆ H1  and   b âˆˆ H2

Since H1 and H2 are subgroups.
Therefore,

    a âˆˆ H1  ,  b âˆˆ H1
â‡¢  ab-1 âˆˆ H1 

and

   a âˆˆ H2 ,  b âˆˆ H2
â‡¢ ab-1 âˆˆ H2

Thus,

   ab-1 âˆˆ H1    and     ab-1 âˆˆ H2
â‡¢ ab-1 âˆˆ H1 âˆ© H2

Hence, H1 âˆ©  H2 is a subgroup of G and that is our theorem i.e. The intersection of two subgroups of a group is again a subgroup.

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