Question

Show that every finite group G is isomorphic to a permutation group.​

Answer 1

Answer:

In group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group G is isomorphic to a subgroup of the symmetric group acting on G.[1] This can be understood as an example of the group action of G on the elements of G.[2]

A permutation of a set G is any bijective function taking G onto G. The set of all permutations of G forms a group under function composition, called the symmetric group on G, and written as Sym(G).[3]

Answer 2

Answer: Theorem A group of permutations is isomorphic to every finite group. One established link is that we can take into account a group G's multiplication (Cayley) table. The elements of the group are permuted in each row. This refers to a symmetric group's subgroup.

Step-by-step explanation: Every group G is isomorphic to a subgroup of the symmetric group acting on G, according to the group theory principle known as Cayley's theorem, which bears Arthur Cayley's name. [1] This can be interpreted as an illustration of group action of G on G's constituent elements. [2]

Any bijective function mapping a set G onto a set G is a permutation of that set. The symmetric group on G, also known as the set of all permutations of G, is a function composition group made up of the set of all permutations of G. (G). [3].

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