0

If G is a group, and g is an element of G, what is the difference between the following two notations o(G) and o(g)?

wirdis
  • 3
  • My guess would be that $o(G)$ is how many elements are in the group and $o(g)$ is the smallest positive integer, $n$, such that $g^n=e$. – J126 Apr 14 '14 at 17:19
  • 1
    The group has an order and so has every element of the group. – drhab Apr 14 '14 at 17:19

2 Answers2

2

Typically $$o(g)=\min\{n\in\mathbb{N}:\,g^n=e\}$$ is the order of the element $g$, whereas $$o(G)=\sum_{x\in G}{1}$$ is the number of elements in the group $G$ as a whole.

1

$o(g)$ is the order of the element, which is the size of the cyclic subgroup generated by the element. Although the notation $|G|$ is more typical, $o(G)$ almost certainly means the order of the group. In general $o(g)$ divides $o(G)$ for any $g \in G$.

user2566092
  • 26,142