Question

D3 a. What is the definition of the determinant of a square matrix over a field: Hint, it is related to permutation groups. b. What is the definition of a normal subgroup of a group? c. Show that any subgroup S of index 2 in a group G (that is, having just two left cosets S, aS), is a normal subgroup. Hint: when is a left coset aS equal to a right coset Sa? It is related to partitions. (An important example: the alternating group An of even permutations is a normal subgroup of the symmetric group of all permutations on n letters Sn.)

Answer 1

a one n is 6 of Definition of Determinant The determinant of a square matrix'n' of Order sum of no terms, for each permutation the set { 1,..., n} where each term ogno Alci... Anon". The entries A16 Anon are one taken one from the each row a and each column and permutation mbolically determinant of A Sgno Alb,... Anon 0 bi 14 Sgns IS the sign of sy TAL b Normal subgroup А subgroup H of a group G is normal if a Ha Ha & at G. We write HAG. © Given s is subgroup of G of index 2. SA G. To prove ΟΥ gs = sg & gEG then If GES true. this is So, assume ges cosets Index Note that left partion G into two disjoint sets ( since the is 2) since, 9€ s there are es are two different left cosets ce-identity of) a an d gs partions G in two two similarly right cosets distoint sets.

These disjoint right cosets are se and Sg gs : Gies = GIS - Gise Sg sa G proved 1 more I hope it will help you to understand Normal sub group clearly. Please comment if any query ! Please Upvote!