Intersection of two subgroups of a group is again a subgroup
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.
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.
Let H1 and H2 be any two subgroups of G.
H1 ∩ H2 ≠ ∅
Since at least the identity element ‘e’ is common to both H1 and H2 .
In order to prove that H1 ∩ H2 is a subgroup, it is sufficient to prove that
a ∈ H1 ∩ H2 , b ∈ H1 ∩ H2 ⇢ a b-1 ∈ H1 ∩ H2
a ∈ H1 ∩ H2 ⇢ a ∈ H1 and a ∈ H2 b ∈ H1 ∩ H2 ⇢ b ∈ H1 and b ∈ H2
Since H1 and H2 are subgroups.
a ∈ H1 , b ∈ H1 ⇢ ab-1 ∈ H1
a ∈ H2 , b ∈ H2 ⇢ ab-1 ∈ H2
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.