...is a subgroup of Gi, and f(G\) is a subgroup of G2. Quotient groups and normal subgroups Definition. A subgroup H of a group G is said to be a normal subgroup if aH = Ha for all a in G. If H is a normal subgroup, the cosets form a group with the operation...

...condition for the right cosets of H in G to yield the same partition as the left cosets is that for each ae G aH = Ha DEFINITION OF NORMAL SUBGROUPS A subgroup,...ae G. Note that aH = Ha for all ae G if and only if <rlha e H for all he H and for all ae G. Hence another definition of normal subgroups is: A subgroup,...

...Therefore, G is hypoabelian if and only if A and B are hypoabelian. A" -subgroups and hypocentral groups A subgroup H of a group G is said to be a .ff -subgroup of G if H = [H,G\. Theorem 2.8 [18] In any group G, there exists a unique maximal K -subgroup...

...the future we shall simply speak of the semidirect product HK N. Characteristic and Fully-Invariant Subgroups A subgroup H of a group G is said to be fully-invariant in G if H* < H for all ae End G, and characteristic in G if H* < H for all ae Aut G....

...Note that R(x) — R(y) => x = eR(x) = eR(y) — y. D DEFINITION 2.1 A normalized transversal T to a subgroup H of a group G is said to be a loop transversal if the right loop (T, *, ||, 1) of Proposition 2.2 is a (two-sided) loop, or in other...