Isomorphism between Additive Group Modulo 16 and Multiplicative Group Modulo 17
Theorem
Let $\struct {\Z_{16}, +}$ denote the additive group of integers modulo $16$.
Let $\struct {\Z'_{17}, \times}$ denote the multiplicative group of reduced residues modulo $17$.
Let $\phi: \struct {\Z_{16}, +} \to \struct {\Z'_{17}, \times}$ be the mapping defined as:
- $\forall \eqclass k {16} \in \struct {\Z_{16}, +}: \map \phi {\eqclass k {16} } = \eqclass {3^k} {17}$
Then $\phi$ is a group isomorphism.
Proof
Let $\eqclass x {16}, \eqclass y {16} \in \struct {\Z_{16}, +}$.
Then:
\(\ds \map \phi {\eqclass x {16} } \times \map \phi {\eqclass y {16} }\) | \(=\) | \(\ds \map \phi {x + 16 m_1} \times \map \phi {y + 16 m_2}\) | Definition of Residue Class: for some representative $m_1, m_2 \in \Z$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 3 \uparrow \paren {x + 16 m_1} \times 3 \uparrow \paren {y + 16 m_2}\) | using Knuth uparrow notation $3 \uparrow k := 3^k$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 3 \uparrow \paren {x + 16 m_1 + y + 16 m_2}\) | Product of Powers | |||||||||||
\(\ds \) | \(=\) | \(\ds 3 \uparrow \paren {\paren {x + y} + 16 \paren {m_1 + m_2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 \uparrow \paren {\eqclass {x + y} {16} }\) | Definition of Residue Class and Definition of Modulo Addition | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \phi {\eqclass x {16} + \eqclass y {16} }\) | Definition of $\phi$ |
Thus it is seen that $\phi$ is a group homomorphism.
$\Box$
It remains to be seen that $\phi$ is a bijection.
Because $17$ is prime:
$\forall x \in \Z, 1 \le x < 17: x \perp 17$
where $\perp$ denotes coprimality.
Thus by definition of multiplicative group of reduced residues modulo $17$:
- $\order {\struct {\Z'_{17}, \times} } = 16$
where $\order {\, \cdot \,}$ denotes the order of a group.
Similarly, by definition of additive group of integers modulo $16$:
- $\order {\struct {\Z_{16}, +} } = 16$
So:
- $\order {\struct {\Z'_{17}, \times} } = \order {\struct {\Z_{16}, +} }$
which is a necessary condition for group isomorphism.
$\Box$
Now we have:
\(\ds 16\) | \(\equiv\) | \(\ds 0\) | \(\ds \pmod {16}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \phi {\eqclass {16} {16} }\) | \(=\) | \(\ds \map \phi {\eqclass 0 {16} }\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \phi {\eqclass {16} {16} }\) | \(=\) | \(\ds \eqclass 1 {17}\) | Group Homomorphism Preserves Identity | ||||||||||
\(\text {(1)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds 3^{16}\) | \(\equiv\) | \(\ds 1\) | \(\ds \pmod {17}\) | Definition of $\phi$ |
Now let $\eqclass x {16}, \eqclass y {16} \in \Z_{16}$ such that $\map \phi {\eqclass x {16} } = \map \phi {\eqclass y {16} }$.
We have:
\(\ds \map \phi {\eqclass x {16} }\) | \(=\) | \(\ds \map \phi {\eqclass y {16} }\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall m_1, m_2 \in \Z: \, \) | \(\ds \map \phi {x + 16 m_1}\) | \(=\) | \(\ds \map \phi {y + 16 m_2}\) | Definition of Residue Class | |||||||||
\(\ds \leadsto \ \ \) | \(\ds 3 \uparrow \paren {x + 16 m_1}\) | \(=\) | \(\ds 3 \uparrow \paren {y + 16 m_2}\) | Definition of $\phi$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds 3^x \paren {3^{16} }^{m_1}\) | \(=\) | \(\ds 3^y \paren {3^{16} }^{m_2}\) | Product of Powers, Power of Power | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds 3^x \times 1^{m_1}\) | \(=\) | \(\ds 3^y \times 1^{m_2}\) | as $3^{16} \equiv 1 \pmod {17}$ from $(1)$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds 3^x\) | \(=\) | \(\ds 3^y\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds y\) |
Thus $\phi$ is an injection.
From Equivalence of Mappings between Finite Sets of Same Cardinality it follows that $\phi$ is a bijection.
$\Box$
Thus $\phi$ is a bijective group homomorphism.
Hence the result by definition of group isomorphism.
$\blacksquare$
Sources
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Group Homomorphism and Isomorphism: $\S 62 \beta$