This first of two volumes covers basic concepts, characters, arithematical properties of characters, products of characters, induced characters and representations, projective representations, Clifford theory, Brauer's induction theorems, faithful characters, the existence of

normal subgroups, and sums of degrees of irreducible characters.

Even the naive translation of

normal subgroups to normal Hopf subalgebras is problematic.

Giving answer to Herstiens's problem, Laffey proved that A(G) = 1 provided G has no non-trivial abelian

normal subgroups [8].

05) while the levels of the single-branch stenosis and

normal subgroups were similar (P greater than 0.

Let H be minimal among

normal subgroups of G with order divisible by r.

Given a prime p, a group is called residually p if the intersection of its p-power index

normal subgroups is trivial, and called virtually residually p if it has a finite index subgroup that is residually p.

Thomas: A complete study of the lattices of fuzzy congruences and

normal subgroups, Information Sciences, 82(1995), 197-218.

In the case of

normal subgroups, one can see even more from the colored group table.

Topics covered include counting of subgroups and proof of the main counting theorems, regular p-groups and regularity criteria, p-groups of maximal class and their characterizations, characters of p-groups, p-groups with large Schur multiplier and commutator subgroups, (p--1)- admissible Hall chains in

normal subgroups, powerful p-groups, automorphisms of p-groups, p-groups that have nonnormal subgroups that are all cyclic, and Alberin's problem of abelian subgroups of small index.

Having studied groups, subgroups,

normal subgroups, and factor groups, it is time to develop a library of examples.

Q], being restrictions of characters in [Mathematical Expression Omitted] to

normal subgroups of [g.

His topics include the Schur-Zassenhaus theorem: a bit of history and motivation, abelian and minimal

normal subgroups,

normal subgroups with abelian sylow subgroups, and groups with specific classes of subgroups complemented.