# How many non-isomorphic groups are there in order 8?

## How many non-isomorphic groups are there in order 8?

four non-isomorphic groups
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. You may list any four of these.

How many non-isomorphic groups are there that have order 6?

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.

What are the non Abelian groups of order less than 8 up to isomorphism?

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It’s often said that there are only two nonabelian groups of order 8 up to isomorphism, one is the quaternion group, the other given by the relations a4=1, b2=1 and bab−1=a3.

### How many groups are there of order 12 up to isomorphism?

So there are two abelian groups of order 12, up to isomorphism, Z2 × Z2 × Z3 and Z4 × Z3.

How many different non isomorphic Abelian groups Order 8 are there?

Now proceed… Conclusion: There are 5 groups of order 8 in which 3 are abelian and 2 are nonabelian.

How many Abelian groups up to isomorphism are there of order 6?

2 groups
Order 6 (2 groups: 1 abelian, 1 nonabelian)

## How many different 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
4 2 2
5 5 1
6 2 1 ⋅ 3 1 1
7 7 1

How many Abelian groups are there of order 6 up to isomorphism?

How many non Abelian group of order 8 are there?

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two non
Looking back over our work, we see that up to isomorphism, there are five groups of order 8 (the first three are abelian, the last two non-abelian): Z/8Z, Z/4Z × Z/2Z, Z/2Z × Z/2Z × Z/2Z, D4, Q.

### How many groups of order 8 are there?

5 groups
It turns out that up to isomorphism, there are exactly 5 groups of order 8.

How many non-Abelian group of order 12 are there?

3 non-abelian groups
We conclude that in addition to the two abelian groups Z12 and Z2 × Z6, there are 3 non-abelian groups of order 12, A4, Dic3 ≃ Q12 and D6.

How many abelian groups of order 16 are there up to isomorphism?

14 groups
There are 14 groups of order 16 up to isomorphism.