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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.

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    $\begingroup$ 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. $\endgroup$ Commented Jul 8, 2022 at 5:31

1 Answer 1

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$H\le G$ . 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 $|G|<\infty$

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  • $\begingroup$ Where did "Then number of distinct cosets $i_G(H) = [G:H]$" come from? Why is the the number of distinct cosets? $\endgroup$
    – Oran
    Commented Jul 8, 2022 at 19:40
  • $\begingroup$ $H\le G$ 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. $\endgroup$ Commented Jul 9, 2022 at 2:45

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