3 Suppose that G is an abelian group Let p be a prime that d

(3) Suppose that G is an abelian group. Let p be a prime that divides IGI (A) If G is a cyclic group, show that G contains an element of order p (B) Show that if G has no proper subgroups (i.e., if SS G, then S 11 or S G), then G contains an element of order p (B) Now, suppose G contains a non-proper subgroup. Suppose further that for every group H of order HI GI we have that if p divides H then H contains an element of order p. Show that G contains an element of order p. (Suggestion: Consider the non-proper subgroup H of G and the quotient group G/H. Apply the assumption and Part (A).) (C1) Next, suppose G is a nonabelian group. If a proper subgroup of G has order divisible by p, then by induction we know that G has an element of order p Thus, we may assume no proper subgroup of G has order divisible by p. Under this assumption, show that if H is a proper subgroup of G then p divides G/H C2 (Still assuming that G is non-abelian having no proper subgroup with order is divisible by p.) Write down the Class Equation. Use Part C1 show that the center Z(G) G and conclude that we have a contradiction and that G must have an element of order p

Solution

(B)

Let G be an abelian group

Let p be a prime that divides |G|

Suppose there is a non-trivial subgroup |H|<|G|

If p divides |H| or p divides |G/H|

By induction that, H has an elelemtn of order p which is also of order p in G

Then |G| contains an element of order p.