# Normal Subgroup

(redirected from Normal subgroups)

## normal subgroup

[′nȯr·məl ′səb‚grüp]
(mathematics)
A subgroup N of a group G where every expression g -1 ng is in N for every g in G and every n in N. Also known as invariant subgroup; normal divisor.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## Normal Subgroup

(also normal divisor of a group, invariant subgroup), a fundamental concept of group theory, which was introduced by E. Galois. A normal subgroup of a group G is a subgroup H for which gH = Hg for arbitrary element g of group G.

References in periodicals archive ?
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).
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.-]|.

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