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In mathematics, the term permutation representation of a (typically finite) group can refer to either of two closely related notions: a representation of as a group of permutations, or as a group of permutation matrices. The term also refers to the combination of the two.
In mathematics, a permutation of a set can mean one of two different things: ... This gives the most compact representation of arbitrary permutations, ...
The nontrivial one-dimensional representation arises through the groups grading: The action is multiplication by the sign of the permutation of the group element. Every finite group has such a representation since it is a subgroup of a cyclic group by its regular action.
A permutation group is a subgroup of a symmetric group; that is, its elements are permutations of a given set. It is thus a subset of a symmetric group that is closed under composition of permutations, contains the identity permutation, and contains the inverse permutation of each of its elements. [2]
The regular representation of G, for a given field K, is the linear representation made by taking this permutation representation as a set of basis vectors of a vector space over K. The significance is that while the permutation representation doesn't decompose – it is transitive – the regular representation in general breaks up into ...
Permutation of four elements with 1 fixed point and 1 3-cycle. In mathematics, the cycles of a permutation π of a finite set S correspond bijectively to the orbits of the subgroup generated by π acting on S. These orbits are subsets of S that can be written as { c 1, ..., c n}, such that π (c i) = c i + 1 for i = 1, ..., n − 1, and π (c n ...
The identity is its minimum, and the permutation formed by reversing the identity is its maximum. If a permutation were assigned to each inversion set using the element-based definition, the resulting order of permutations would be that of a Cayley graph, where an edge corresponds to the swapping of two elements on consecutive places. This ...
The representation theory of groups is a part of mathematics which examines how groups act on given structures. Here the focus is in particular on operations of groups on vector spaces. Nevertheless, groups acting on other groups or on sets are also considered. For more details, please refer to the section on permutation representations.