Please help with subgroup question Please show all work Assu

Assume that G is a group with a proper subgroup H such that|G| = 33and|H| > 10. Find|G|, |H|,and [G: H], A group G has subgroups of orders 4 and 10, and |G|

Solution

9).

IGI=33 and IHI>10

=> IGI/IHI <3.3

=> [G:H] <3.3 (denition of the index [G:H ])

Now, we know [IGI,IHI] <3.3 and IGI=33 that gives IHI >10

10).

From Lagrange’s theorem we know that both 4 and 10 divide|G|.

From theorem 12.3, this means that |G| is divisible by 20 (Least Common Factor), the least common multiple of (4,10). We’re also given that|G|<50.

There are only two multiples of 20 that are less than 50, so we know that

|G| {20,40}.