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Full-Text Articles in Algebra
Impartial Avoidance And Achievement Games For Generating Symmetric And Alternating Groups, Bret J. Benesh, Dana C. Ernst, Nándor Sieben
Impartial Avoidance And Achievement Games For Generating Symmetric And Alternating Groups, Bret J. Benesh, Dana C. Ernst, Nándor Sieben
Mathematics Faculty Publications
Anderson and Harary introduced two impartial games on finite groups. Both games are played by two players who alternately select previously unselected elements of a finite group. The first player who builds a generating set from the jointly-selected elements wins the first game. The first player who cannot select an element without building a generating set loses the second game. We determine the nim-numbers, and therefore the outcomes, of these games for symmetric and alternating groups.
Impartial Avoidance Games For Generating Finite Groups, Bret J. Benesh, Dana C. Ernst, Nándor Sieben
Impartial Avoidance Games For Generating Finite Groups, Bret J. Benesh, Dana C. Ernst, Nándor Sieben
Mathematics Faculty Publications
We study an impartial avoidance game introduced by Anderson and Harary. The game is played by two players who alternately select previously unselected elements of a finite group. The first player who cannot select an element without making the set of jointly-selected elements into a generating set for the group loses the game. We develop criteria on the maximal subgroups that determine the nim-numbers of these games and use our criteria to study our game for several families of groups, including nilpotent, sporadic, and symmetric groups.
The Probabilistic Zeta Function, Bret Benesh
The Probabilistic Zeta Function, Bret Benesh
Mathematics Faculty Publications
This paper is a summary of results on the PG(s) function, which is the reciprocal of the probabilistic zeta function for finite groups. This function gives the probability that s randomly chosen elements generate a group G, and information about the structure of the group G is embedded in it.
A Classification Of Certain Maximal Subgroups Of Alternating Groups, Bret Benesh
A Classification Of Certain Maximal Subgroups Of Alternating Groups, Bret Benesh
Mathematics Faculty Publications
This paper addresses and extension of Problem 12.82 of the Kourovka notebook, which asks for all ordered pairs (n,m) such that the symmetric groups Sn embeds in Sm as a maximal subgroup. Problem 12.82 was answered in a previous paper by the author and Benjamin Newton. In this paper, we will consider the extension problem where we allow either or both of the groups from the ordered pair to be an alternating group.
A Classification Of Certain Maximal Subgroups Of Symmetric Groups, Benjamin Newton, Bret Benesh
A Classification Of Certain Maximal Subgroups Of Symmetric Groups, Benjamin Newton, Bret Benesh
Mathematics Faculty Publications
Problem 12.82 of the Kourovka Notebook asks for all ordered pairs (n,m) such that the symmetric group Sn embeds in Sm as a maximal subgroup. One family of such pairs is obtained when m=n+1. Kalužnin and Klin [L.A. Kalužnin, M.H. Klin, Certain maximal subgroups of symmetric and alternating groups, Math. Sb. 87 (1972) 91–121] and Halberstadt [E. Halberstadt, On certain maximal subgroups of symmetric or alternating groups, Math. Z. 151 (1976) 117–125] provided an additional infinite family. This paper answers the Kourovka question by producing a third infinite family of ordered …