Question #109079
p and Q are subgroups of a group G and o(P) and o(Q) relatively prime prove that p intersection Q=e,e belonge to G
1
Expert's answer
2020-04-13T17:56:41-0400

Let P and QP \ and \ Q are two subgroup of GG and O(P) and O(Q)O(P) \ and \ O(Q) are relatively prime ,i,e,gcd(O(P),O(Q))=1.gcd(O(P),O(Q))=1.

Claim: PQ={e}P\cap Q=\{ e\} ,where ee is the identity element of GG .

Since, intersection of two subgroup is again a subgroup.

 PQ\therefore \ P\cap Q is a subgroup of G.G.

But PQP as well as  PQQP\cap Q\sube P \ \text{as well as } \ P\cap Q \sube Q , so PQP\cap Q is a subgroup of PP as well as QQ by definition of subgroup.

Therefore by Lagrange's theorem, i,e, order of every subgroup divided the order of the group .We get

O(PQ)O(P) and O(PQ)O(Q)O(P\cap Q) \mid O(P) \ and \ O(P\cap Q) \mid O(Q)

    O(PQ)gcd(O(P),O(Q)).\implies O(P\cap Q)\mid gcd(O(P),O(Q)).

    O(PQ)1\implies O(P\cap Q)\mid 1

    O(PQ)=1\implies O(P\cap Q)=1 .

Hence,PQ={e}P\cap Q=\{e\} .Since the subgroup of order one in a group is identity element itself.


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!
LATEST TUTORIALS
APPROVED BY CLIENTS