A5 - subgroup

order of a group

$| G |$ denotes the order of a group, which is the number of elements in it

order of an element

for $a \in G$ , its order $| a | = n$ such that it is the smallest integer for which $a^{n} = e$

consider $U (15) = {1, 2, 4, 7, 8, 11, 13, 14}$ under multiplication modulo 15
the order of the group: $| U (15) | = 8$
the order of element: $| 7 | = 4$ as $7^{4} = 1$
a tip to calculate the order: $13 = - 2 \mod 15$ , so $13^{2} = (- 2)^{2} = 4$ , $13^{3} = - 2 \cdot 4 = - 8$ , $13^{4} = - 2 \cdot - 8 = 1$

subgroup

if a subset $H$ of a group $G$ is itself a group under the operation of $G$ , $H$ is a subgroup of $G$ , denoted by: $H \leq G$