[x14] Let be the quaternion group ([bd8c]). Prove that can’t be written as a semidirect product in any nontrivial fashion.
[x15] Let be a group of order , where and have no common factor. If contains exactly one subgroup of order and exactly one subgroup of order , prove that is the direct product of and .
[x16] Prove that .
[x17] Let be the quaternion group ([bd8c]). Prove that .
[x18] Let be the set of all matrices in of the form , . Check that is a subgroup of , and prove that it is a semidirect product of (additive group) by . Is it a direct product of these two groups?
[x19] Find the automorphism groups of and .
[x19a]Let , where and are finite groups, and let be an element of with and . Denote the order of an element by
(a) Show that for some divisor of .
(b) When acts trivially on , show that
(c) Let with . Let and let . Show that , , and .
(d) Suppose that , where is cyclic of order and that, for some generator of ,
Show inductively that, for ,
( copies of ). Deduce that has order (hence in this case).
(e) Suppose that , where is commutative, is cyclic of order , and the generator of acts on by sending each element to its inverse. Show that has order no matter what is (in particular, is independent of ).
[x19b]Let be the semidirect of its subgroups and , and let
(centralizer of in ). Show that
Let be the homomorphism giving the action of on (by conjugation). Show that if is commutative, then
and if and are commutative, then
[x19c]A homomorphism of groups is normal if is a normal subgroup of . The cokernel of a normal homomorphism is defined to be . Show that, if in the following commutative diagram, the blue sequences are exact and the homomorphisms are normal, then the red sequence exists and is exact:
[x19d]Let and be subgroups of , and assume that normalizes , i.e., for all . Let denote the action of on , . Show that
is a homomorphism with image .
[x19e]Let and be subgroups of a group . Show that is the semidirect product of and if and only if there exists a homomorphism whose restriction to is the identity map and whose kernel is .
