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Is S_n a subgroup of S_n + 1? (n = 1, 2, ...). What is the largest with sea element in S_5 can
have?
Solution
(1) Indeed Sn may be regarded as a subgroup of Sn+1 (in several ways).
For example , let H be the subgroup of all g in Sn+1 that leave 1 fixed. (We regard Sn+1 as the
group of all permutationsof the n+1 letters 1,2,3...n+1).
Any element of H gets naturally identified with a permutation of the n symbols {2,3,...,n+1),
hence may be regarded as an element of Sn.
In other words Sn+1 contains (several, as there is nothing special about 1, we could have taken
any symbol from 1,2,3..to n+1) copy of Sn as a subgroup.
(b) The largest order that an element can have in S5 is 6., corresponding to the element
(12)(345). (This is the optimal decomposition of a permutation in S5 which gives the maximum
order)

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Is S_n a subgroup of S_n + 1 (n = 1, 2, ...). What is the large.pdf

  • 1. Is S_n a subgroup of S_n + 1? (n = 1, 2, ...). What is the largest with sea element in S_5 can have? Solution (1) Indeed Sn may be regarded as a subgroup of Sn+1 (in several ways). For example , let H be the subgroup of all g in Sn+1 that leave 1 fixed. (We regard Sn+1 as the group of all permutationsof the n+1 letters 1,2,3...n+1). Any element of H gets naturally identified with a permutation of the n symbols {2,3,...,n+1), hence may be regarded as an element of Sn. In other words Sn+1 contains (several, as there is nothing special about 1, we could have taken any symbol from 1,2,3..to n+1) copy of Sn as a subgroup. (b) The largest order that an element can have in S5 is 6., corresponding to the element (12)(345). (This is the optimal decomposition of a permutation in S5 which gives the maximum order)