# Basic Aspects in Group Theory. Part 7: Properties of the Sylow Decomposition.

In this story we follow-up what we have started here about looking into the so-called Sylow decomposition of a commutative, finite generated group with all elements having finite order. Our ultimate goal is to obtain a uniqueness result in the decomposition of groups into as small as possible non-trivial factors, but in order to accomplish this, we first need to extract some useful properties of the Sylow decomposition.

First, for convenience, let us restate the main proposition of our last story:

Remember, `S_p`

denotes the subgroup of all elements that have an order of power of `p`

. The direct sum is made over all prime numbers `p`

.

This result brings us really close to our goal of showing uniqueness of the decomposition into as small as possible subgroups. We will attain at this remarkable result in the next story.

Thank you for reading!