Dick said:Pick an element g of G that is not e and consider the subgroup generated by g.
catherinenanc said:Ok, this may sound stupid, but, how do I know that <g>=G?
To prove that G is cyclic, we need to show that there exists an element g in G such that every element in G can be written as a power of g. This can be done by showing that the order of g is equal to the order of the group G.
A prime p order group is a group in which the number of elements is a prime number p. This means that the group has exactly p elements and every element in the group has an order that is a factor of p.
The order of a group is equal to the number of elements it contains. This can be determined by counting the number of elements in the group or by using a formula, such as Lagrange's theorem, which states that the order of any subgroup must divide the order of the group.
Yes, a group can be cyclic even if it is not of prime p order. For a group to be cyclic, there just needs to exist an element g in the group such that every element can be written as a power of g. This does not depend on the order of the group.
Proving that a group is cyclic can provide insight into the structure of the group and can also make certain calculations and proofs easier. Additionally, cyclic groups have many important applications in mathematics and other fields, making it a useful concept to understand.