Let G be a finite group. A. let p be a prime such that p divides |G|, the order

Question

Let G be a finite group. A. let p be a prime such that p divides |G|, the order of G . prove that G contains a subgroup of order p.b. if G contains a subgroup of order p, must p divide |G| ? c. up to isomorphism, what are the grops of order 6? If you list two groups that are isomorphic, that will be considered a wrong answer. d. does A _4 have a subgroup of order 6? Does A _4 have a normal subgroups or order 4? e. if G is abelian and m divides |G| where m is a positive integer, must G contain a subgroup of order m? f. if G is abelian and m divides |G| where m is a positive integer, must G contain an element of order m?

Explanation / Answer

As converse of Lagrange theorem does not hold good for A4 alternating groups, but holds good for other finite groups,

here G is a group and p divides order of G

Hence there is an element a such that a^p =e

The elements {e, a, a^2,...a^(p-1)} with p elements form a subgroup of G with order p

------------------------------------------------------

Yes. Order of a subgroup divides order of group.

Hence p divides n order of G

--------------------------------------------------------

Related Questions

Navigate

• Browse (All)
• Browse L
• Subjects

---

---

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