How many abelian groups are there in order 25?

There are exactly two abelian groups of order 25, up to isomorphism.

Understanding Groups of Order 25

The question of how many abelian groups exist for a given order is directly linked to the properties of that order. For an order like 25, which is a perfect square of a prime number ($25 = 5^2$), the classification becomes straightforward. It is a known mathematical fact that any group whose order is the square of a prime number (like $p^2$) must be abelian. This significantly simplifies the problem, as we only need to identify the distinct group structures of order 25, all of which will inherently be abelian.

The Fundamental Theorem of Finite Abelian Groups

The Fundamental Theorem of Finite Abelian Groups is the cornerstone for classifying finite abelian groups. It states that every finite abelian group can be uniquely expressed as a direct product of cyclic groups of prime-power order.
For a group of order $p^n$ (where $p$ is a prime and $n$ is a positive integer), the number of non-isomorphic abelian groups is given by the partition function $P(n)$. A partition of an integer $n$ is a way of writing $n$ as a sum of positive integers, where the order of the summands does not matter.

In our case, the order is 25, which can be written as $5^2$. Here, the prime $p=5$ and the exponent $n=2$. We need to find the partitions of $n=2$:

• 2: This corresponds to a cyclic group of order $p^2$.
• 1 + 1: This corresponds to a direct product of two cyclic groups, each of order $p$.

The Two Abelian Groups of Order 25

Based on the partitions of 2, the two distinct (non-isomorphic) abelian groups of order 25 are:

1. The cyclic group of order 25: This group is isomorphic to $\mathbb{Z}{25}$ (or often denoted as $C{25}$). This is the group of integers modulo 25 under addition. It is generated by a single element, for example, 1.

2. The direct product of two cyclic groups of order 5: This group is isomorphic to $\mathbb{Z}_5 \times \mathbb{Z}_5$ (or $C_5 \times C_5$). This group consists of pairs of elements $(a, b)$ where $a, b \in {0, 1, 2, 3, 4}$, and the operation is addition modulo 5 component-wise. This group is not cyclic; no single element can generate all 25 elements.

These are the only two possible structures for a group of order 25, and as established, both are abelian.