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# In $(\mathbb{Z}_{25}^*,\otimes)$ the set H={1,6,11,16,21} is a subgroup. Find the equivalence classes of the congruence mod H relation.

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DEFINITIONS

An $\textbf{operation}$ on $A$ is a function whose domain is $A\times A$

$\textbf{Commutative property}$: $\forall a,b\in A, a\ast b=b\ast a$

$\textbf{Closure property}$: $\forall a,b\in A, a\ast b\in A$

$\textbf{Associative property}$: $\forall a,b,c\in A, (a\ast b)\ast c=a\ast (b\ast c)$

$\textbf{Identity element}$: element $e$ such that $\forall a\in A, a\ast e=e\ast a=a$

$\textbf{Inverse}$: Let $a\in A$. The inverse of $a$ is the element $b$ such that $a\ast b=b\ast a=e$

$\textbf{Group}$: $(G,\ast)$ is a group if

$\bullet$ $G$ is closed under $\ast$

$\bullet$ $\ast$ is associative

$\bullet$ $\ast$ has an identity element

$\bullet$ Every element of $G$ has an inverse element in $G$

$H\subseteq G$ is a $\textbf{subgroup}$ of $(G, \ast)$ if $(H, \ast)$ is also a group.

$\textbf{Congruence modulo a subgroup}$: $a$ is congruent to $b$ modulo $H$ if $a\ast b^{-1}\in H$.

The $\textbf{equivalence class}$ of $a$ is the set of all elements that are relation to $a$. Notation: $[a]_R$

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