Find the index of (1) in the group S 3 , using the notation of Example 10.7

Find the index of (µ1) in the group S₃, using the notation of Example 10.7

Data from 10.7 Example

Table 10.8 again shows Table 8.8 for the symmetric group S₃ on three letters. Let H be the sub group (µ ₁) = { p₀ , µ _i} of S₃ . Find the partitions of S₃ into left co sets of H, and the partition into right cosets of H.

 

Transcribed Image Text:

For the partition into left cosets, we have H = {po, M1}, p₁H = {P₁P0, P₁M₁} = {P₁, M3}, ριμι} {P₂P0, P₂M₁} = {P₂, M₂}. P₂ H The partition into right cosets is = H = {po, μ1}, Hp₁ Hp₂ = {pop1, 1P1} = {P₁, M₂}, μερι} = {POP2, M1P₂} = {P₂, M3}.