# Notes - Groups HT23, HCF and LCM

### Flashcards

Let $m, n$ be non-zero integers. In the context of the infinite cyclic group $(\mathbb Z, +)$, what could you define that acts very much like the highest common factor and lowest common multiple?

\[\begin{aligned}
\langle h \rangle &= \langle m, n \rangle \\\\
\langle l \rangle &= \langle m \rangle \cap \langle n \rangle
\end{aligned}\]

What’s the notation for “$x$ divides $y$”?

\[x \vert y\]

Using the $x \vert y$ notation, how would you write that $x$ is even?

\[2\vert x\]

What is Bézout’s lemma?

Let $m, n$ be integers with highest common factor (gcd) $h$. Then there exists integers $a, b$ s.t. $an + bm = h$.

### Proofs

Let $m, n$ be integers. By defining (in the context of the infinite cyclic group $(\mathbb Z, +)$)

\[\langle h \rangle = \langle m, n \rangle\]
prove Bézout’s identity,

\[\exists a, b\in \mathbb Z \text{ s.t. } an + bm = h\]

Todo.

Let $m,n \in \mathbb Z^+$ be positive coprime integers. Prove that

\[C_n \times C_m \cong C_{mn}\]

Todo.