If G is a finite group with |G|&lt;180 and G has subgroups of orders 10, 18 and 30 then the order of G is:1) 602) 903) 304) 80

"If G is a finite group with |G|<180..." solved and explained

College
Algebra
Abstract algebra

Junaid Ayala

Answered question

2022-03-01

If G is a finite group with

| G | < 180

and G has subgroups of orders 10, 18 and 30 then the order of G is:
1) 60
2) 90
3) 30
4) 80

Answer & Explanation

Jocelyn Harwood

Beginner2022-03-02Added 8 answers

Solution:
Let G be the finite group and G has subgroups of order 10, 18 and 30.
By Lagrange's theorem,
The order of each subgroup of a finite group is a divisor of the order of the group.

O (G) = L . C . M

(order of subgroups)

O (G) = L . C . M (10, 18, 30)

∴ L . C . M (10, 18, 30) = 2 \times 3 \times 5 \times 3

= 90

\begin{array}{cc} 2 & 10, 18, 30 \\ 3 & 5, 9, 15 \\ 5 & 5, 3, 5 \\ 1, 3, 1 \end{array}

Then,

O (G) = 90

(2) option is correct

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