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Intersection of two subgroups of a group is again a subgroup

Last Updated : 05 Mar, 2021
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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 ∩  H2  ≠  ∅

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

   a ∈ H1 ∩ H2 ,  b ∈ H1 ∩ H2
⇢ a b-1 ∈ H1 ∩ H2

Now,

   a ∈ H1 ∩ H2  
⇢ a ∈ H1  and   a ∈ H2
   b ∈ H1 ∩ H2
⇢ b ∈ H1  and   b ∈ H2

Since H1 and H2 are subgroups.
Therefore,

    a ∈ H1  ,  b ∈ H1
⇢  ab-1 ∈ H1 

and                     

   a ∈ H2 ,  b ∈ H2
⇢ ab-1 ∈ H2

Thus,       

   ab-1 ∈ H1    and     ab-1 ∈ H2
⇢ ab-1 ∈ H1 ∩ H2

Hence, H1 ∩  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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