Determine all distinct right and left cosets of H11inU24 2 A

Determine all distinct right and left cosets of H=(11)inU_24 2. Assume that G is a group with a subgroup H such that|G|13 and [G:H]> 4.Find |G|,|H|, and [G:H].

Solution

Solution : 2 )

We know that |H| divides |G| and [G:H] = |G|/|H|.
So, we let |H| = m and |G| = mn for some positive integers m and n
(and thus [G:H] = n).

If [G:H] = 5, then |G| = 5 |H|. Then, |H| = 14 ==> |G| = 70. If |H| > 14, then |G| > 70 which contradicts |G| < 71.

If [G:H] 6, then |G| 6 |H|. Then, |H| 14 ==> |G| 84, which contradicts |G| < 71.

Hence, there is only one solution : |G| = 70, |H| = 14, and [G:H] = 5.