List the elements of a subgroup

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The question is to list the elements of $H \subset \mathbb{Z}_{1610}$ if $H = \langle 1035\rangle$. I know that there are 14 elements in the set, I am not sure how to find the elements.

Is the correct answer $115, 345, 575, 1035, 1265, 1495$?

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1 Answer

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Clearly $H$ is a cyclic group generated by 1035.

You might have read that if $H=\langle a \rangle$ then elements of $H$ are powers of $a$

i.e. $H=\{ a^0,a^1,a^2,a^3,......\} $

Here you know that you have 14 elements (I haven't checked and assuming you're correct) so

$H=\{ a^0,a^1,a^2,...,a^{13}\} $ would give you a list of unique elements if you calculate any further powers you would get one of these as in that case $a^{14}=a^0$ and $a^{14+i}=a^{i(\mod 14)}$

Note that $H$ is an additive subgroup, so $a^i=i\cdot a$ (adding $a$ $i$ times).

Also it is a subgroup of $\mathbb{Z}_{1610}$ so addition would take modulo 1610

Thus $a^i(\mod 1610)$ where $i=0,1,2,...13$ would give you your list.

Note $a^0=e$ the identity element of the group (here 0).

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