Let P and Q are two subgroup of G and O(P) and O(Q) are relatively prime ,i,e,gcd(O(P),O(Q))=1.
Claim: P∩Q={e} ,where e is the identity element of G .
Since, intersection of two subgroup is again a subgroup.
∴ P∩Q is a subgroup of G.
But P∩Q⊆P as well as P∩Q⊆Q , so P∩Q is a subgroup of P as well as Q by definition of subgroup.
Therefore by Lagrange's theorem, i,e, order of every subgroup divided the order of the group .We get
O(P∩Q)∣O(P) and O(P∩Q)∣O(Q)
⟹O(P∩Q)∣gcd(O(P),O(Q)).
⟹O(P∩Q)∣1
⟹O(P∩Q)=1 .
Hence,P∩Q={e} .Since the subgroup of order one in a group is identity element itself.
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