Action Of Symmetric Group S5 On The Cosets Of Some Of Its Non-Maximal Subgroups

Abstract

The action of the symmetric group S5 on the cosets of its non-maximal subgroups is an aspect that seems to have received less attention for sometime. Most studies have infact looked on the action of symmetric group S5 on the cosets of its maximal subgroups leaving many properties about its action on non-maximal subgroups unknown. This project has therefore looked at the action of the symmetric group S5, on the cosets of some of its non-maximal subgroups. Namely; the cyclic groups Cn, n = 3 ; 4 ; 5 and 6, the symmetric group S3, the direct products C2C3, C2C4 and C2C5. Corresponding to each action, the permutation representations, polynomial formulas, subdegrees and rank has been computed. To get the cycle index formulas, we rst come up with the disjoint cycle structures by considering intersection of conjugacy classes with the respective subgroups.Then the cycle index formulas are derived by summing the product monomials. For the subdegrees and ranks we use Burnside matrix that enhances computation of the groups marks,the marks are used to solve system of linear equations obtained from a table of marks, that gives us the rank and subdegrees of a given subgroup . When S5 acts on its non-maximal subgroups we have in most cases two unique subdegrees When S5 act on H where H is any cyclic subgroup Cn n = 3 or 5 the subdegrees are 4(1) and 􀀀5!􀀀4n n2(n) and the rank is (4n2􀀀4n+5!)n2 : Also when S5 act on H where H is any cyclic subgroup Cn n = 4 or 6, the symetric group S3 and the direct product C2C3. The subdegrees are 2(1) and 􀀀5!􀀀2n n2 (n) and the rank is (2n2􀀀2n+5!) n2 :

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APA

LYDIAH, O (2021). Action Of Symmetric Group S5 On The Cosets Of Some Of Its Non-Maximal Subgroups. Afribary. Retrieved from https://afribary.com/works/action-of-symmetric-group-s5-on-the-cosets-of-some-of-its-non-maximal-subgroups

MLA 8th

LYDIAH, OMORI "Action Of Symmetric Group S5 On The Cosets Of Some Of Its Non-Maximal Subgroups" Afribary. Afribary, 31 May. 2021, https://afribary.com/works/action-of-symmetric-group-s5-on-the-cosets-of-some-of-its-non-maximal-subgroups. Accessed 28 May. 2024.

MLA7

LYDIAH, OMORI . "Action Of Symmetric Group S5 On The Cosets Of Some Of Its Non-Maximal Subgroups". Afribary, Afribary, 31 May. 2021. Web. 28 May. 2024. < https://afribary.com/works/action-of-symmetric-group-s5-on-the-cosets-of-some-of-its-non-maximal-subgroups >.

Chicago

LYDIAH, OMORI . "Action Of Symmetric Group S5 On The Cosets Of Some Of Its Non-Maximal Subgroups" Afribary (2021). Accessed May 28, 2024. https://afribary.com/works/action-of-symmetric-group-s5-on-the-cosets-of-some-of-its-non-maximal-subgroups