Abelian groups as inner mapping groups of loops

The question of which Abelian groups can be the inner mapping group of a loop has been considered by Niemenmaa, Kepka and others. We give a construction which shows that many finite Abelian groups can be the inner mapping group of a loop.

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On finite loops whose inner mapping groups are Abelian II

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700038491 ◽

2005 ◽

Vol 71 (3) ◽

pp. 487-492

Author(s):

Markku Niemenmaa

Keyword(s):

Abelian Group ◽

Direct Product ◽

Prime Power ◽

Abelian Groups ◽

Finite Abelian Group ◽

Prime Power Order ◽

Finite Abelian Groups ◽

Cyclic Groups ◽

Inner Mapping Group ◽

Mapping Group

If the inner mapping group of a loop is a finite Abelian group, then the loop is centrally nilpotent. We first investigate the structure of those finite Abelian groups which are not isomorphic to inner mapping groups of loops and after this we show that if the inner mapping group of a loop is isomorphic to the direct product of two cyclic groups of the same odd prime power order pn, then our loop is centrally nilpotent of class at most n + 1.

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On finite loops whose inner mapping groups are Abelian

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700020529 ◽

2002 ◽

Vol 65 (3) ◽

pp. 477-484 ◽

Cited By ~ 4

Author(s):

Markku Niemenmaa

Keyword(s):

Solvable Group ◽

Abelian Groups ◽

Solvable Groups ◽

Special Structure ◽

Finite Abelian Groups ◽

Multiplication Group ◽

Nonassociative Algebras ◽

Inner Mapping Group ◽

Mapping Group

Loops are nonassociative algebras which can be investigated by using their multiplication groups and inner mapping groups If the inner mapping group of a loop is finite and Abelian, then the multiplication group is a solvable group. It is clear that not all finite Abelian groups can occur as inner mapping groups of loops. In this paper we show that certain finite Abelian groups with a special structure are not isomorphic to inner mapping groups of finite loops. We use our results and show how to construct solvable groups which are not isomorphic to multiplication groups of loops.

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Counting subgroups of fixed order in finite abelian groups

Journal of Discrete Mathematical Sciences and Cryptography ◽

10.1080/09720529.2020.1833440 ◽

2021 ◽

Vol 24 (1) ◽

pp. 263-276

Author(s):

Fikreab Solomon Admasu ◽

Amit Sehgal

Keyword(s):

Abelian Groups ◽

Finite Abelian Groups ◽

Fixed Order

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EXAMPLES OF FINITE-DIMENSIONAL POINTED HOPF ALGEBRAS IN CHARACTERISTIC 2

Glasgow Mathematical Journal ◽

10.1017/s0017089520000579 ◽

2020 ◽

pp. 1-14

Author(s):

NICOLÁS ANDRUSKIEWITSCH ◽

DIRCEU BAGIO ◽

SARADIA DELLA FLORA ◽

DAIANA FLÔRES

Keyword(s):

Hopf Algebras ◽

Abelian Groups ◽

Vector Spaces ◽

Group Algebras ◽

Finite Abelian Groups ◽

Finite Dimensional ◽

Pointed Hopf Algebras ◽

Characteristic 2 ◽

Nichols Algebras ◽

Kirillov Dimension

Abstract We present new examples of finite-dimensional Nichols algebras over fields of characteristic 2 from braided vector spaces that are not of diagonal type, admit realizations as Yetter–Drinfeld modules over finite abelian groups, and are analogous to Nichols algebras of finite Gelfand–Kirillov dimension in characteristic 0. New finite-dimensional pointed Hopf algebras over fields of characteristic 2 are obtained by bosonization with group algebras of suitable finite abelian groups.

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