Permutation group: Difference between revisions
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In mathematical [[group theory]], the set of [[permutation]]s on a set of objects form a [[group theory|group]], is called a '''permutation group''', with composition as the group operation. For example, let <math>A</math> denote a finite set of <math>n</math> distinct objects, and let <math>S_A</math> denote the set of permutations of the elements of <math>A</math>. The criteria of [[associativity]] and the existence of inverses are obvious from the definition of permutations as [[bijections]] from <math>A</math> to itself. The existence of an [[identity (algebra)|identity]] is slightly more difficult to establish, but we can define an identity mapping <math>i(a)=a,\forall a\in A</math>, and it is clear that this mapping is both a permutation and an identity.<ref>Fraleigh J B (2003) ''A First Course in Abstract Algebra'' ISBN 0321156080, section 8</ref> | |||
Since all permutation groups over sets with the same number of elements are [[isomorphism (algebra)|isomorphic]], and since [[abstract algebra]] is generally only concerned with groups, [[ring theory|rings]], [[field (algebra)|fields]] ''et cetera'' [[up to]] isomorphism, the generic permutation group over <math>n</math> elements is simply denoted <math>S_n</math>. | Since all permutation groups over sets with the same number of elements are [[isomorphism (algebra)|isomorphic]], and since [[abstract algebra]] is generally only concerned with groups, [[ring theory|rings]], [[field (algebra)|fields]] ''et cetera'' [[up to]] isomorphism, the generic permutation group over <math>n</math> elements is simply denoted <math>S_n</math>. | ||
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A [[symmetry group]] is in general a [[subgroup]] of the permutation group over the number of [[vertex|vertices]] in the figure in question, since all [[affine transformation]]s of a figure are permutations of its vertices, although the converse does not in general hold. | A [[symmetry group]] is in general a [[subgroup]] of the permutation group over the number of [[vertex|vertices]] in the figure in question, since all [[affine transformation]]s of a figure are permutations of its vertices, although the converse does not in general hold. | ||
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Latest revision as of 16:01, 2 October 2024
In mathematical group theory, the set of permutations on a set of objects form a group, is called a permutation group, with composition as the group operation. For example, let denote a finite set of distinct objects, and let denote the set of permutations of the elements of . The criteria of associativity and the existence of inverses are obvious from the definition of permutations as bijections from to itself. The existence of an identity is slightly more difficult to establish, but we can define an identity mapping , and it is clear that this mapping is both a permutation and an identity.[1]
Since all permutation groups over sets with the same number of elements are isomorphic, and since abstract algebra is generally only concerned with groups, rings, fields et cetera up to isomorphism, the generic permutation group over elements is simply denoted .
A symmetry group is in general a subgroup of the permutation group over the number of vertices in the figure in question, since all affine transformations of a figure are permutations of its vertices, although the converse does not in general hold.
References
- ↑ Fraleigh J B (2003) A First Course in Abstract Algebra ISBN 0321156080, section 8