Thus [[OMEGA].sub.2](A)B is the product of two elementary abelian

normal subgroups. Since p is odd, we see that [[OMEGA].sub.2](A)B has exponent p, which is a contradiction.

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.

The plasma renalase levels of the multi-branch and two-branch stenosis subgroups were significantly lower than that of the subgroup with normal coronary angiography outcomes (P less than 0.05) while the levels of the single-branch stenosis and

normal subgroups were similar (P greater than 0.05).

Let H be minimal among

normal subgroups of G with order divisible by r.

P(G) = {P([G.sub.1]) [union] P([G.sub.2]), [*.sub.1], [*.sub.2]} is said to be a neutrosophic normal subbigroup of [B.sub.N](G) if P(G) is a neutrosophic subbigroup and both P([G.sub.1]) and P([G.sub.2]) are

normal subgroups of B([G.sub.1]) and B([G.sub.2]) respectively.

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.

Let [H.sub.1] and [H.sub.2] be

normal subgroups of G.

Let [H.sup.+] and [H.sup.-] subsets of [[??].sub.p] [??] S such that [H.sup.+] = [[??].sub.p] [??] {e} = {(u, e); u [member of] [[??].sub.p]} and [H.sup.-] = Ker(v) = {(u, s); v(u, s) = e}, then Both [H.sup.+] and [H.sub.-] are

normal subgroups of [[??].sub.p] [??] S and S [congruent to] [[??].sub.p] [??] S/[H.sup.+] [congruent to] [[??].sub.p] [??] S/[H.sup.-], with p = |[H.sup.+]| = |[H.sup.-]|.