A3 - important groups

additive group

integers with addition modulo n, $(Z, +)$
eg: $Z_{4} = {0, 1, 2, 3}$
- here, $(3 + 1) \mod 4 = 1$
- this gives it closure
identity $: e = 0$ , as $i + 0 = 0 + i = i$ for $i \in Z$
inverse $: a^{'} = n - a$ , as $a + (n - a) = n (\mod n) = 0 = e$
it is an abelian group

abelian groups

a group $G$ is said to be abelian if for $a, b \in G :$

a * b = b * a

unitary group

drawing a cayley table of $(Z_{6}, \times)$ and cancelling out all the rows and columns without an identity as shown below:

the remaining members, $1$ and $5$ , are the only invertible elements
these form the special unitary group $U_{6} = 1, 5$

special unitary group

$U (n)$ it is a group of integers that are relatively prime to $n$ under multiplication modulo $n$

U (n) = {m | 0 < m < n; gcd (m, n) = 1; m, n \in Z}

general linear group

general linear group

$G L (n, F)$ the group of all invertible square matrices with entries from $F$ under matrix multiplicaition

invertible $⟹ det \neq 0$
if the entries are integers, then $F = Z_{p}$ for $det \neq 0$ , eg: $G L (2, Z_{p})$ with identity $I = diag (1, 1)$
they are non-abelian as matrix multiplication is not commutative