Question

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

Short answer

The index of $<a^4>$in the group $<a>$ is 2.

Step-by-step solution

To determine the order of  H

Let us see the definition of an index of a subgroup.

Let H be a subgroup of group G. Then, the index of a subgroup H in a group Gis defined as the number of distinct left cosets of H in Gor the number of distinct right cosets of H in G.

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

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

It is given that $o\left(a\right)=30$.

Let $K=<a>$ be the group generated by a and $H=<a^4>$ be the subgroup of K generated by $a^4$.

Then,

role="math" localid="1652331505494" $H=\left\{\begin{array}{ll}{a}^4,a^8,a^{12},a^{16},a^{20},& a^{24},a^{28},a^{32}=a^2,a^{40}=a^{10},\\ a^{52}=a^{22},a^{56}=a^{26},& a^{60}=0\end{array}\right.\left.\begin{array}{r}{a}^{44}=a^{14},a^{48}=a^{18},\\ \end{array}\right\}$

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

To find the index of  H

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

Then by Lagrange’s Theorem,

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

$=\frac{30}{15}$

$=2.....(\because o(a)=\hspace{0.17em}30$and $a\in G)$

$\therefore$index of$H=2$

Hence, the index of $<a^4>$ in the group $<a>$ is 2.