1. prove no two of C_8, C_4 X C_2, C_2 X C_2 X C_2 are isomorphic.C_n is cyclic group of order n.
Give, with justification, a group of order 8 that is not isomorphic to any of those groups.
2. Prove that if m and n are coprime then C_m X C_n is cyclic.
if m and n are not coprime. can it be cyclic?
3. H is a normal subgroup of a finite group G. Which of the followings are true?
(i) if G is cyclic then H and G/H are cyclic.
(ii) If H and G/H are cyclic then G is cyclic.
(iii) If G is abelian then H and G/H are abelian
(iv) If H and G/H are abelian then G is abelian.
4. find all homomorphisms between C_11 to C_14
5. H is a subgroup of G and H not equal to G. Show that there is an element of G which does not belong to any subgroup of the form gHg^(-1) for g belongs to G
False. Let and its 3-element subgroup.(ii) If H and G/H are cyclic then G is cyclic.
Yes. Define the natural map then is a homomorphic image of an abelian group so it must be abelian. Or you can say since .(iii) If G is abelian then H and G/H are abelian
False. Use the same example.(iv) If H and G/H are abelian then G is abelian.
Now it would be nice if this union can generalize, i.e. if is any group then for implies for some . But sadly this does not work. Consider to be an -tuple . Now let . Then they unionize to . But no one groups contains them all.