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**i) Every C-Cycle, C>1, can be written as a product of transpositions. Every permutation of a finite number n>1 of elements can be expressed as a product of transpositions.**

ii) (Cayley's representation theorem for groups) For every group G is isomorphic to a group of permutations A [G]=n then G is isomorphic to a group of permutations on objects.

ii) (Cayley's representation theorem for groups) For every group G is isomorphic to a group of permutations A [G]=n then G is isomorphic to a group of permutations on objects.

**i) a) (Proof of C >1) Every C cycle can be represented in the form (c_2,c_1), (c_3,c_1),...(c_i,c_1),....,(c_c,c_1). Hence this defines the mapping from c_1 to c_c, thus in general it suffices to say that 1 <= i < C. As there is a mapping from c_1 to c_c such that every other number in the group remains unchanged, then i=1, thus that 1=n. Hence 1 < C.**

b) (Proof of n > 1) Hence from a), any C-Cycle with C>1 can be written as a product of transpositions. Thus as permutations can be represented as C cycles on a arbitary set then n > 1 is satisfied by part a) #.

ii) (Proof of one to one correpondence) Let the set A be defined as (p_1, p_2, .... , p_n,...) Hence, let A undergo the mapping p_1 -> g_1 such that the image of A under the mapping is: (p_1g_1, ..., p_ng_n,....). Thus there is an equivalence between p_n ~ (p_1g_1,....., p_ng_n).

(Proof of Isomorphism) For an isomorphism to be acting from A to the permutation set then associotivity must hold. Let p_k = (p_1g_1,....., p_1g_k), Hence if G is isomorphic to the permutation of G then p_k(p_n * c) = (p_k * p_n)c. p_k * p_n = ([p_1,.....,p_k],[p_1g_1,.....,p_kg_k,...]) * (([p_1,.....,p_n],[p_1g_1,.....,p_ng_n,....]). As the permutation group has the same binary operation as the group itself. Then there is an isomorphism between the group G and the group of permutations on the group.

b) (Proof of n > 1) Hence from a), any C-Cycle with C>1 can be written as a product of transpositions. Thus as permutations can be represented as C cycles on a arbitary set then n > 1 is satisfied by part a) #.

ii) (Proof of one to one correpondence) Let the set A be defined as (p_1, p_2, .... , p_n,...) Hence, let A undergo the mapping p_1 -> g_1 such that the image of A under the mapping is: (p_1g_1, ..., p_ng_n,....). Thus there is an equivalence between p_n ~ (p_1g_1,....., p_ng_n).

(Proof of Isomorphism) For an isomorphism to be acting from A to the permutation set then associotivity must hold. Let p_k = (p_1g_1,....., p_1g_k), Hence if G is isomorphic to the permutation of G then p_k(p_n * c) = (p_k * p_n)c. p_k * p_n = ([p_1,.....,p_k],[p_1g_1,.....,p_kg_k,...]) * (([p_1,.....,p_n],[p_1g_1,.....,p_ng_n,....]). As the permutation group has the same binary operation as the group itself. Then there is an isomorphism between the group G and the group of permutations on the group.

I'm having trouble understanding the concept of an isomorphism, the book I'm working from defines it as a special kind of equavlence which acts on the relation between two groups, whether that is binary, etc. So the group may have different contents but if the operation holds then there is an isomorphism. Is this correct or am I doing something wrong?