## How many Abelian group of orders are there?

By the fundamental theorem for finite abelian groups the number of abelian groups of order n=pn11… pnkk is the product of the partition numbers of ni. Note that the partition number of 2 is 2 and the partition number of 4 is 5. Since 106=26⋅56 such an n therefore exists.

### How many abelian groups of order 216 are there?

9

Statistics at a glance

Quantity | Value |
---|---|

Number of abelian groups | 9 |

Number of nilpotent groups | 25 |

Number of solvable groups | 177 |

Number of simple groups | 0 |

**How many groups of order 256 are there?**

56092 groups

gap> SmallGroupsInformation(256); There are 56092 groups of order 256.

**How many Abelian group of order 108 are there?**

two Abelian groups

Page 210 Problem 6 Show that there are two Abelian groups of order 108 that have exactly one subgroup of order 3.

## How many groups are there in order 25?

Table of number of distinct groups of order n

Order n | Prime factorization of n | Number of groups |
---|---|---|

23 | 23 1 | 1 |

24 | 2 3 ⋅ 3 1 | 15 |

25 | 5 2 | 2 |

26 | 2 1 ⋅ 13 1 | 2 |

### Is a group of order 25 abelian?

Every group of order 25 = 52 is abelian, so there are two possibilities for K. This gives rise to two groups: G3: the direct product Z/11Z × Z/5Z × Z/5Z G4: one nonabelian semidirect product Z/11Z Χ (Z/5Z × Z/5Z).

**How many abelian groups up to isomorphism are there of order 360?**

six different abelian groups

There are six different abelian groups (up to isomorphism) of order 360. A group G is decomposable if it is isomorphic to a direct product of two proper nontrivial subgroups. Otherwise G is indecomposable. The finite indecomposable abelian groups are exactly the cyclic groups with order a power of a prime.

**How many abelian groups of order are there up to isomorphism?**

6 Abelian groups

11.38 Note that up to isomorphism, there are 6 Abelian groups of order 72, namely, G1 × G2 for G1 ∈ {ZZ8,ZZ2 × ZZ4,ZZ2 × ZZ2 × ZZ2}, and G2 ∈ {ZZ9,ZZ3 × ZZ3}.

## Are there any abelian groups of prime order?

, both of which are Abelian. If , and the number grows very rapidly as the power increases. Every group of prime order is cyclic, since Lagrange’s theorem implies that the cyclic subgroup generated by any of its non-identity elements is the whole group.

### How many abelian groups are in bijection up to isomorphism?

Furthermore, abelian groups of order 16 = 24, up to isomorphism, are in bijection with partitions of 4, and abelian groups of order 9 = 32 are in bijection with partitions of 2. Thus, there are 5 2 = 10 abelian groups of order 144 and they are Z

**Are there any groups of order that are solvable?**

The cyclic numbers ( are relatively prime . is solvable. A theorem of William Burnside, proved using group characters, states that every group of order is divisible by fewer than three distinct primes. By the Feit–Thompson theorem, which has a long and complicated proof, every group of order is odd. For every positive integer are solvable.

**Are there infinitely many simple groups of the same order?**

To see this for any particular order is usually not difficult (for example, there is, up to isomorphism, one non-solvable group and ) but the proof of this for all orders uses the classification of finite simple groups . simple groups of that order, and there are infinitely many pairs of non-isomorphic simple groups of the same order.