The Number of Groups of Order n
Main Concept
A group is a set endowed with a group operation, a function mapping pairs of group elements back to the group. That is, if x and y are elements of a group G and f is the group operation, then fx,y is also a group element, usually denoted x∘y. The group operation must satisfy the following properties:
(Associativity)
For all x, y, z ∈G,
x∘y∘z =x∘y∘z.
(Identity)
∃ e∈G s.t. for all x ∈G,
x∘e = e∘x = x.
(Inverse)
For all x ∈G, ∃ y∈G s.t.
x∘y = y∘x = e.
The number of elements of a group is called its order.
Two groups are called isomorphic if there is a bijective mapping between the groups that preserves the group operation. The "number of groups of order n" refers to the number of isomorphism classes of groups of order n.
The graph below displays the number of groups of order n for various different orders. Check "Logarithmic Scale" to choose a logarithmic scale on the vertical axis, otherwise a linear scale is used on the vertical axis. Select an item from the "Highlight Orders" list to highlight group orders of special interest. Use the sliders or type in the boxes (and then click outside them) to change the minimum and maximum group orders.
Note: As you change the slider for the maximum or minimum order, the other value may change as well to accommodate the new value.
NothingPrimesPrime PowerPrimes SquaredPrimes CubedSquare FreeCube FreeDivisible by Exactly Two PrimesDivisible by at Most Two PrimesDivisible by Exactly Three PrimesDivisible by at Most Three Primes
Minimum order:
Maximum order:
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