5 votes

Let (G,*) be a group such that O(G) = 8, where O(G) denotes the order of the group. Which of the following is True ?

- There exist no element a in G whose order is 6.
- There exist an element a in G whose order is 4.
- There exist more then one element in G whose order is 1
- None of these

6 votes

Best answer

Let $G$ be a finite group of order $k$

then every element of $G$ will have order $m$ such that $k \ \text{mod } m = 0$

**A** is true because $8 \ \text{mod } 6 = 2$

**B **is false because $8 \ \text{mod } 4 = 0$ but it is not necessary that there must exist an element with order 4.

for example Elementary abelian group:E8 is of order 8 but does not contain any element of order 4

**C **is false because there must exist only 1 identity element in a group