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Say an element $g$ in a group has order $28$. How do I find the order of say $g^8$?
Guest said:Say an element $g$ in a group has order $28$. How do I find the order of say $g^8$?
Thanks. I wonder whether there's a systematic way of working this out if one has to find $g^i$ for all $2 \le i \le 27$?I like Serena said:Hi Guest,
We are looking for the lowest $k$ such that $(g^{8})^k = 1$.
And we know that $28$ is the lowest such that $g^{28} = 1$.
That means we're looking for the lowest $k$ such that $8k$ is a multiple of $28$.
That is:
$$k = \frac{\text{lcm}(28,8)}{8}$$
Guest said:Thanks. I wonder whether there's a systematic way of working this out if one has to find $g^i$ for all $2 \le i \le 27$?
The order of an element in a group is the number of times that the element must be multiplied by itself to get the identity element of the group.
The order of an element can be calculated by finding the smallest positive integer n such that an = e, where a is the element and e is the identity element.
The order of an element in a group can tell us important information about the structure and properties of the group. For example, the order of an element can determine the number of subgroups in the group and can be used to determine if the group is cyclic or abelian.
Yes, the order of an element can be infinite in groups that have an infinite number of elements. In these cases, the element can be multiplied by itself an infinite number of times and still not equal the identity element.
The order of an element can determine the pattern of its powers. For example, if the order of an element is n, then its powers will cycle through n distinct values before repeating. Additionally, the order of an element can tell us if every element in the group can be expressed as a power of that element.