# Let $|a| = n$. How many left cosets of $\langle a^k \rangle$ in $\langle a \rangle$ are there?

a) Let $$|a| = 40$$. How many left cosets of $$\langle a^6 \rangle$$ in $$\langle a \rangle$$ are there?

b) Let $$|a| = n$$. How many left cosets of $$\langle a^k \rangle$$ in $$\langle a \rangle$$ are there?

For part a, I managed to list out all of the elements of $$\langle a^6 \rangle$$. Taking the identity $$\epsilon$$ and doing $$\epsilon \langle a^6 \rangle$$, I get the first coset and there turned out to be $$20$$ elements. So by the property that all cosets of a subgroup have the same cardinality, two cosets are either equal or disjoint, and each element appears in exactly one coset, I concluded that there are $$2$$ left cosets of $$\langle a^6 \rangle$$ in $$\langle a \rangle$$.

I am not sure how to show part b. Is there any connection from part a to b? Any help would be great, thanks.

Also, I have not yet learned Lagrange's theorem.

• Part a) was likely intended to be a hint. Work out similarly many $(n,k)$ pairs, and see how the land lies! Suitable piece of theory were likely covered in the book/lecture notes. Jul 8, 2022 at 5:31

$$H\le G$$ . Then number of distinct cosets (left/right) $$i_G(H) =[G:H]$$

Let $$G=\langle a\rangle$$ and $$H=\langle a^k\rangle$$

Then $$|G|=40$$ and $$|H|=\frac{40}{\gcd(k, 40) }$$

\begin{align}i_G(H) =\frac{|G|}{|H|}&=\frac{40}{\frac{40}{\gcd(k, 40) }}\\&=\gcd(k,40)\end{align}

Note:

1. $$H\le G$$ means $$H$$ is a subgroup of $$G$$

2. $$i_G(H) =\frac{|G|}{|H|}$$ as $$|G|<\infty$$

• Where did "Then number of distinct cosets $i_G(H) = [G:H]$" come from? Why is the the number of distinct cosets?
– Oran
Jul 8, 2022 at 19:40
• $H\le G$ and $a, b\in G, a\sim b$ iff $ab^{-1}\in H$ . Prove that it's an equivalence relation. $[a]=aH$ . Then $G=\bigcup_a aH$ . Use $|aH|=|bH|=|H|$ . That's it. Jul 9, 2022 at 2:45