21. Prove or disprove: Every nonabelian group of order divisible by 6 contains a subgroup of...
Prove or disprove There exists nonabelian group G (order of group G is 219 3 x 73)
(a) Let be a cyclic group of order . Prove that for every divisor of there is a subgroup of having order . (b) Characterize all factor groups of
(6)(20 points) (a) Let G be a cyclic group of order n. Prove that for every divisor dofn there is a subgroup of Ghaving order d. (b) Characterize all factor groups of Z70.
(6)(20 points) (a) Let G be a cyclic group of order n. Prove that for every divisor d of n there is a subgroup of G having order d. (b) Characterize all factor groups of Z70 -
Show that every group of order 55 has both a subgroup of order 5 and a subgroup of order 11.
(6 points) Let G be a group of order 35. Show that every non-trivial subgroup of G is cyclic.
#7 7 Prove or disprove: If H is a normal subgroup of G such that H and G/H are abelian, then G is abelian. If G is cyclic, prove that G/H must also be cyclic. 8.
(3) (9 marks) Show that every group of order 55 has both a subgroup of order 5 and a subgroup of order 11.
(a) Let G be a cyclic group of order n. Prove that fo every divisor d of n there is a subgroup of G having order d. (b) Characterize all factor groups of Z70.
Prove or disprove: The relation "is-a-normal-subgroup-of" is a transitive relation. (please do not solve using facts about the index)