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Cayley's theorem
Cayley's Theorem: A group G is cyclic if and only if every non-identity element in G has order equal to the order of G. Order of a group G: The orde...
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Cayley's Theorem: A group G is cyclic if and only if every non-identity element in G has order equal to the order of G. Order of a group G: The orde...
Cayley's Theorem:
A group G is cyclic if and only if every non-identity element in G has order equal to the order of G.
Order of a group G:
The order of a group G is the smallest positive integer n such that G contains an element of order n.
Automorphism of a group G:
An automorphism of a group G is a function f: G -> G that is not the identity function.
Cayley's Theorem says that:
A group G is cyclic if and only if it has a finite number of automorphisms.
The number of automorphisms of a group G is equal to the order of G.
A group G is cyclic if and only if it has a unique minimal normal subgroup of order equal to the order of G.
Examples:
The cyclic group of order 5 has exactly 5 automorphisms, which are the permutations of the cyclic group.
The cyclic group of order 6 has 3 automorphisms, which are the cyclic group itself, the anti-cyclic group, and the identity permutation.
The cyclic group of order 7 has only 1 automorphism, which is the identity permutation.
The group of order 12 has 2 automorphisms, which are the identity permutation and the permutation that sends each element to its next cyclic order element