# Why a group of order 9 is abelian?

## Why a group of order 9 is abelian?

If an element c has order 9, then 1, c, c^2 c^8 is the whole group, and is obviously Abelian. If not, then every element except 1 has order 3, x^3 = 1 and x^2 =/= 1. Likewise, these elements are distinct from each other; so that’s the whole group.

What is a subgroup of an abelian group?

Every subgroup of an abelian group is normal, so each subgroup gives rise to a quotient group. Subgroups, quotients, and direct sums of abelian groups are again abelian. The finite simple abelian groups are exactly the cyclic groups of prime order.

How many groups are there of order 9?

There are, up to isomorphism, two possibilities for a group of order 9….Groups of order 9.

Group GAP ID (second part) Defining feature
cyclic group:Z9 1 unique cyclic group of order 9
elementary abelian group:E9 2 unique elementary abelian group of order 9; also a direct product of two copies of cyclic group:Z3.

### Is an Abelian group solvable?

Every abelian group is solvable. For, if G is abelian, then G = H0 ⊇ H1 = {e} is a solvable series for G. Every nilpotent group is solvable.

Is a subgroup of an abelian group abelian?

Yes, subgroups of abelian groups are indeed abelian, and your thought process has the right idea. Showing this is pretty easy. Take an abelian group G with subgroup H.

Are all subgroups of abelian groups abelian?

#### Are all groups of order 9 cyclic?

Both of these are abelian groups and, in particular are abelian of prime power order….Groups of order 9.

Group GAP ID (second part) Defining feature
cyclic group:Z9 1 unique cyclic group of order 9
elementary abelian group:E9 2 unique elementary abelian group of order 9; also a direct product of two copies of cyclic group:Z3.

How many abelian groups are there of order 6?

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