18.703 Modern Algebra, Presentations and Groups of small order
نویسنده
چکیده
i Given any word w, the reduced word w associated to w is any word obtained from w by reduction, such that wi cannot be reduced any further. Given two words w1 and w2 of A, the concatenation of w1 and w2 is the word w = w1w2. The empty word is denoted e. The set of all reduced words is denoted FA. With product defined as the reduced concatenation, this set becomes a group, called the free group with generators A.
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18.703 Modern Algebra, Permutation groups
Proof. (5.2) implies that the set of permutations is closed under com position of functions. We check the three axioms for a group. We already proved that composition of functions is associative. Let i : S −→ S be the identity function from S to S. Let f be a permutation of S. Clearly f ◦ i = i ◦ f = f . Thus i acts as an identity. Let f be a permutation of S. Then the inverse g of f is a perm...
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