> Q:

> How many groups are there of order 6?

> A:

> There are two groups of order 6.

> To determine this, we consider the possible structures of a group of order 6. Since 6 = 2 × 3, where both 2 and 3 are prime numbers, the groups of order 6 can either be cyclic or a direct product of smaller cyclic groups.

> Cyclic Group of Order 6: There is always one cyclic group for any given order. The cyclic group of order 6 is denoted as 𝐶₆ or ℤ₆, which is generated by an element of order 6.

> Non-Cyclic Group: The other possibility is a non-cyclic group. By Sylow's theorems, the number of elements of order 2 must divide 3 and be congruent to 1 modulo 2, and the number of elements of order 3 must divide 2 and be congruent to 1 modulo 3. This means there must be one Sylow 2-subgroup and one Sylow 3-subgroup. The non-cyclic group of order 6 is the symmetric group 𝑆₃, which is the group of all permutations of three objects. It is non-abelian and can also be thought of as the semi-direct product of 𝐶₂ and 𝐶₃.

> So, there are two groups of order 6: the cyclic group ℤ₆ and the symmetric group 𝑆₃.