The object of study in this paper is the finite groups whose integral group rings have only trivial central units. Prime-power groups and metacyclic groups with this property are characterized. Metacyclic groups are classified according to the central height of the unit groups of their integral group rings.
Let q be an odd prime power and p be an odd prime with gcdðp; qÞ ¼ 1: Let order of q modulo p be f ; gcdð pÀ1 f ; qÞ ¼ 1 and q f ¼ 1 þ pl: Here expressions for all the primitive idempotents in the ring R p n ¼ GF ðqÞ½x=ðx p n À 1Þ; for any positive integer n; are obtained in terms of cyclotomic numbers, provided p does not divide l if nX2: The dimension, generating polynomials and minimum distances of minimal cyclic codes of length p n over GF ðqÞ are also discussed. r 2004 Elsevier Inc. All rights reserved.
We provide algorithms to compute a complete irredundant set of extremely strong Shoda pairs of a finite group G and the set of primitive central idempotents of the rational group algebra Q[G] realized by them. These algorithms are also extended to write new algorithms for computing a complete irredundant set of strong Shoda pairs of G and the set of primitive central idempotents of Q[G] realized by them. Another algorithm to check whether a finite group G is normally monomial or not is also described.
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