The subgroup structure of a group tells us much about the group. For example, th

Question

The subgroup structure of a group tells us much about the group. For example, there are groups that have no non-trivial proper subgroups(sungroups of order larger than 1 that are not the entire group) and these groups have a very simple structure.

(a) Suppose G is a finite group that contains no non-trivial proper subgroups. What can you say about |G|? Prove your answer

(b) What conditions are there on the order of a finite group G that ensure that G contains a non-trivial proper subgroup? Prove your answer

(c) Correctly complete the statement of the following theroem: A group G of order n has a non-trivial proper subgroup if and only if n is __________

This is abstract algebra

Explanation / Answer

a) Let |G| 2 (possibly |G| = ). If G has no proper nontrivial subgroups, then G and e are the only subgroups. Let a G be a nonidentity element. Then the subgroup generated by a cannot be e, so a = G, hence G is cyclic. If |G| = , then G = Z. But Z has nontrivial proper subgroups. Thus |G| < . Suppose |G| = n. Since G is cyclic ,for every divisor d of n,there exist a subgroup H of order d. If d is not 1 or d is not n, then H is a proper, nontrivial subgroup of G. Therefore, the only divisors of n are n and 1, hence n is prime.

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.