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Displaying similar documents to “A note on group algebras of p -primary abelian groups”

On ( σ , τ ) -derivations in prime rings

Mohammad Ashraf, Nadeem-ur-Rehman (2002)

Archivum Mathematicum

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Let R be a 2-torsion free prime ring and let σ , τ be automorphisms of R . For any x , y R , set [ x , y ] σ , τ = x σ ( y ) - τ ( y ) x . Suppose that d is a ( σ , τ ) -derivation defined on R . In the present paper it is shown that ( i ) if R satisfies [ d ( x ) , x ] σ , τ = 0 , then either d = 0 or R is commutative ( i i ) if I is a nonzero ideal of R such that [ d ( x ) , d ( y ) ] = 0 , for all x , y I , and d commutes with both σ and τ , then either d = 0 or R is commutative. ( i i i ) if I is a nonzero ideal of R such that d ( x y ) = d ( y x ) , for all x , y I , and d commutes with τ , then R is commutative. Finally a related result has been obtain...

Making sense of capitulation: reciprocal primes

David Folk (2016)

Acta Arithmetica

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Let ℓ be a rational prime, K be a number field that contains a primitive ℓth root of unity, L an abelian extension of K whose degree over K, [L:K], is divisible by ℓ, a prime ideal of K whose ideal class has order ℓ in the ideal class group of K, and a any generator of the principal ideal . We will call a prime ideal of K ’reciprocal to ’ if its Frobenius element generates G a l ( K ( a ) / K ) for every choice of a . We then show that becomes principal in L if and only if every reciprocal prime is not...

On the Davenport constant and group algebras

Daniel Smertnig (2010)

Colloquium Mathematicae

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For a finite abelian group G and a splitting field K of G, let (G,K) denote the largest integer l ∈ ℕ for which there is a sequence S = g · . . . · g l over G such that ( X g - a ) · . . . · ( X g l - a l ) 0 K [ G ] for all a , . . . , a l K × . If (G) denotes the Davenport constant of G, then there is the straightforward inequality (G) - 1 ≤ (G,K). Equality holds for a variety of groups, and a conjecture of W. Gao et al. states that equality holds for all groups. We offer further groups for which equality holds, but we also give the first examples of groups G for...

On unit group of finite semisimple group algebras of non-metabelian groups up to order 72

Gaurav Mittal, Rajendra Kumar Sharma (2021)

Mathematica Bohemica

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We characterize the unit group of semisimple group algebras 𝔽 q G of some non-metabelian groups, where F q is a field with q = p k elements for p prime and a positive integer k . In particular, we consider all 6 non-metabelian groups of order 48, the only non-metabelian group ( ( C 3 × C 3 ) C 3 ) C 2 of order 54, and 7 non-metabelian groups of order 72. This completes the study of unit groups of semisimple group algebras for groups upto order 72.

A characterization of the linear groups L 2 ( p )

Alireza Khalili Asboei, Ali Iranmanesh (2014)

Czechoslovak Mathematical Journal

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Let G be a finite group and π e ( G ) be the set of element orders of G . Let k π e ( G ) and m k be the number of elements of order k in G . Set nse ( G ) : = { m k : k π e ( G ) } . In fact nse ( G ) is the set of sizes of elements with the same order in G . In this paper, by nse ( G ) and order, we give a new characterization of finite projective special linear groups L 2 ( p ) over a field with p elements, where p is prime. We prove the following theorem: If G is a group such that | G | = | L 2 ( p ) | and nse ( G ) consists of 1 , p 2 - 1 , p ( p + ϵ ) / 2 and some numbers divisible by 2 p , where p is a prime...

On sequences over a finite abelian group with zero-sum subsequences of forbidden lengths

Weidong Gao, Yuanlin Li, Pingping Zhao, Jujuan Zhuang (2016)

Colloquium Mathematicae

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Let G be an additive finite abelian group. For every positive integer ℓ, let d i s c ( G ) be the smallest positive integer t such that each sequence S over G of length |S| ≥ t has a nonempty zero-sum subsequence of length not equal to ℓ. In this paper, we determine d i s c ( G ) for certain finite groups, including cyclic groups, the groups G = C C 2 m and elementary abelian 2-groups. Following Girard, we define disc(G) as the smallest positive integer t such that every sequence S over G with |S| ≥ t has nonempty zero-sum...