Group Theory and Social Groups.

[rmas]

Hi,
I have a question related to Group Theory and its interpretation from a social point of view.
if we suppose, that a group of Humans can be considered as an algebraic structure : a group (G,◦) with a set of elements and a set of axioms like closure, associativity, identity and invertibility.
How can we describe ◦ and the axioms in term of interactions and relations between humans (socially)
I was wondering if someone can give me an idea about this subject, or some pointers if it was already treated.
Thank you.

 

[Rasalhague]

Would this work? It's just something that occurred to me a few weeks ago when I was first learning about groups. Let f stand for "friend", and e for "enemy", and * for "of my". We know that the enemy of my enemy is my friend, and of course the friend of my friend is my friend, and the friend of my enemy is my enemy...

e * e = f
e * f = e
f * f = f
f * e = e

So we have closure. This structure is associative:

f * (e * e) = f * f = f
(f * e) * e = e * e = f

(f * f) * e = f * e = e
f * (f * e) = f * e = e

e * (f * e) = e * e = f
(e * f) * e = e * e = f

f * (e * f) = f * e = e
(f * e) * f = e * f = e

Notice that friend is the identity element, and that the inverse of enemy is enemy because

e * e = f

and the inverse of friend is friend because

f * f = f.

So I reckon that makes a group, and an abelian group at that since we also have commutativity.

 

[Rasalhague]

Aha, I think what I was describing is this, $$C_2$$, the simplest nontrivial group:

http://mathworld.wolfram.com/CyclicGroupC2.html
http://en.wikipedia.org/wiki/List_of_small_groups

Or, if you prefer it in song:



Whether group theory has serious applications in modelling social groups, I don't know...