Learn English – What’s the difference between ‘group’ and ‘grouping’

From Wikipedia

Human societies are characterized by patterns of relationships between individuals sharing a distinctive culture and institutions. Without an article, the term refers either to the entirety of humanity or a contextually specific subset of people.

I would restate one part of the original question as
'"society" (without "the") refers to human society in general'

and, the example should read
"socialization is the process of learning to live in society"

The phrase "the society" is used in areas of study such as anthropology, political science and sociology when referring to specific groups.

Again from the Wikipedia article, discussing a particular group

This nobility organized warriors to protect the society from invasion.

In this case, "the society" is used to limit the scope to the group under discussion, and not all of humanity.

Edited to respond to OP's edit of original question

In "Soldiers protect society" the lack of an article preceding "society" makes it a statement about human society in general. As such it can stand alone without other context.

"Soldiers protect the society" seems taken out of context. It begs the question which society?

(Note: The following example statements are not meant to be historically accurate.)

In general, soldiers protect (human) society.
In ancient Rome, soldiers called centurions protected the (Roman) society.
In ancient Japan, soldiers called samurai protected the (Japanese) society.

Well, when we say community, we refer to a large number of people or entities, who may be affiliated to many smaller or disparate sub-structures.Community almost always refers to a varied and large audience, yet with certain things, which maybe certain interests, opinions, or religion or ethnicity.

On the other hand, group is indeed most often used for a small number of people or other entities, each of which may or may not be large. Also, a group is not as natural a collection as community. A group maybe formed by some of us just now, but a community arises on its own and comes together, and has more naturally common attributes. Similarly, one can be expelled from a group, if he disrespects the rules, but one cannot be banned from a community per se, unless it is something very artificial like a virtual social networking aggregate that is moderated etc.

E.g. we talk about the international community of researchers of a certain field, say Physics, or the international community of institutional investors (each of which is an institution like a bank and consists of millions of people), whereas we say G8- a group of nations.