Question

Let be an element of order 30 in a group $\mathit{G}$.What is the index of$\mathbf{}<a^4>$in the group $<a>\mathbf{?}$

Short answer

The index of in the group is 2.

Step-by-step solution

To determine order of H

Let us see the definition of index of a subgroup,

Let $H$be a subgroup of group $G$.Then the index of a subgroup$H$in a group $G$is defines as the number of distinct left cosets of $H$in $G$or number of distinct right cosets $H$of in $G$.

The index is calculated by dividing the order of group by the order of subgroup, if the orders of group and subgroup are known.

Let $G$be a group and $a\in G$be any element of $G$.

Given that $o\left(a\right)=30$.

Letbe the group generated by $a$andbe the subgroup of $K$generated by $a^4$.

Then,

$$\begin{gathered} H=\left\{a^4, a^8, a^{12}, a^{16}, a^{20}, a^{24}, a^{28}, a^{32}= a^2, a^{36}=a^6, a^{40}= a^{10}, a^{44}=a^{14}, a^{48}=a^{18}, \\ {a}^{52}=a^{22}, a^{56}=a^{26}, a^{60}=e \\ \right\} \end{gathered}$$

Thus, $o\left(H\right)=15$.

To find index of H.

Now, let us find the index of $H$in $K$.

Then by Lagrange’s Theorem,

Index of $H=\frac{o\left(G\right)}{o\left(H\right)}$

$$\begin{gathered} =\frac{30}{15} \\ =2 \end{gathered}$$ …… ( $\because o\left(a\right)=30$and $a\in G$)

$\therefore$index of $H=2$

Hence, the index of in the group is 2.