How many non-isomorphic groups of order 4 are there?
Table of number of distinct groups of order n
| Order n | Prime factorization of n | Number of non-Abelian groups |
|---|---|---|
| 3 | 3 1 | 0 |
| 4 | 2 2 | 0 |
| 5 | 5 1 | 0 |
| 6 | 2 1 ⋅ 3 1 | 1 |
How many non-isomorphic groups of order 6 are there?
In the first part of the question, I showed that every group of order less than 6 is Abelian. In the second part of the question I am asked to show that there are exactly 2 non-isomorphic groups of order 6.
How many non-isomorphic abelian groups of order 64 are there?
… are 11 ways of expressing 6 as a sum of positive integers, and thereby 11 nonisomorphic abelian groups of order 64 = 2 6 .
How many different non-isomorphic groups of order 30 are there?
4 non-isomorphic groups
So these are non-isomorphic groups and there are exactly 4 non-isomorphic groups of order 30. 2.12 #8 Let G be a group of order 231 = 3 × 7 × 11. Let sp be the number of p-Sylow subgroups of G.
How many non-isomorphic abelian groups of order 32 are there?
See classification of finite abelian groups and structure theorem for finitely generated abelian groups. , there are exactly three maximal class groups: dihedral, semidihedral, and generalized quaternion. For order 32, the groups are: dihedral group:D32, semidihedral group:SD32, and generalized quaternion group:Q32.
How many different non-isomorphic groups of order 8 are there?
List four non-isomorphic groups of order 8. There are five groups of order 8, namely Z/8Z, (Z/4Z) × (Z/2Z), (Z/2Z) × (Z/2Z) × (Z/2Z), D4 (or D8) and the quaternion group Q8.
How many non-isomorphic abelian groups of order 1800 are there?
12 structurally different abelian
There are exactly 3 · 2 · 2 = 12 structurally different abelian groups of order n = 1800.
How many abelian groups up to isomorphism are there of order 60?
13
Group counts
| Quantity | Value |
|---|---|
| Total number of groups up to isomorphism | 13 |
| Number of abelian groups (i.e., finite abelian groups) up to isomorphism | 2 |
| Number of nilpotent groups (i.e., finite nilpotent groups) up to isomorphism | 2 |
| Number of solvable groups (i.e., finite solvable groups) up to isomorphism | 12 |
How many non isomorphic abelian groups of order 32 are there?