SQUARE45

Let

n

be a positive integer. The multiplicative group of units modulo

n

, denoted by

(\mathbb{Z}/n\mathbb{Z})^*

, is a finite abelian group whose order is

\phi(n)

. The exponent of this group,

\lambda(n)

, is the smallest positive integer such that

a^{\lambda(n)} \equiv 1 \pmod{n}

for all

a \in (\mathbb{Z}/n\mathbb{Z})^*

. Furthermore, the structure theorem for finite abelian groups dictates that

(\mathbb{Z}/n\mathbb{Z})^*

is isomorphic to the direct product of cyclic groups whose orders are determined by the prime factorization of

n

. Specifically, if

n = p_1^{k_1} p_2^{k_2} \cdots p_r^{k_r}

, then

(\mathbb{Z}/n\mathbb{Z})^* \cong \prod_{i=1}^{r} (\mathbb{Z}/p_i^{k_i}\mathbb{Z})^*

. The exponent

\lambda(n)

is given by the least common multiple of the exponents of the component groups:

\lambda(n) = \operatorname{lcm}(\lambda(p_1^{k_1}), \dots, \lambda(p_r^{k_r}))

. For prime powers

p^k

, the exponent

\lambda(p^k)

is defined as:\n\begin{itemize}\item If

p=2

and

k \ne 1

\lambda(2^k) = 2^{k-2}

.\item If

p

is an odd prime,

\lambda(p^k) = \phi(p^k) = p^{k-1}(p-1)

.\end{itemize}

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