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Chapter 8: 20E (page 246)

A group has fewer than 100elements and subgroups of order of10and25. What is the order of?

Short Answer

Expert verified

The order of G is 50.

Step by step solution

01

Lagrange’s theorem

If K is a subgroup of a finite group G , then the order of K divides the order ofG .

02

Order of G

Let G be the group containing less than 100 elements and H and K be the subgroups of order 10 and 25.

The subgroup H of order 10 has the elements {10,20,30,40,50,60,70,80,90} and the subgroup K of order 25 has the elements {20,50,75} .

The common element in the both the subgroup is 50.

Therefore, the order of G is 50.

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Most popular questions from this chapter

Let $H$ and $K$ be subgroups of a finite group $G$ such that $[G : H] = p$ and $[G : K] = q$ , with $p$ and $q$ distinct primes. Prove that $p q$ divides $[G : H \cap K]$ .

Let N be a cyclic normal subgroup of a group G , and H any subgroup of N . Prove that H is a normal subgroup of G .[Compare Exercise 14]

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8. $H = ⟨ 3 ⟩; G = ℤ_{12}$

If K and N are normal subgroups of a group G , prove that $K \cap N$ is a normal subgroup of G .

In Exercises 7-1 $G$ is a group and $H$ is a subgroup of G. Find the index $[G : H]$ .

11. $H$ is the subgroup generated by $(\begin{matrix} \begin{array}{l} 1 \\ 2 \end{array} & \begin{array}{l} 2 \\ 3 \end{array} & \begin{array}{l} 3 \\ 4 \end{array} & \begin{array}{l} 4 \\ 1 \end{array} \end{matrix}); G = S_{4}$

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