This thread has been dormant for some time, and the class which motivated it is probably by now over, but I've just stumbled across it and can't help but mention an interesting and surprising example of elementary group theory making an appearance in anthropology.
This comes from an appendix, "On the Algebraic Study of Certain Types of Marriage," that Andre Weil wrote for a book of Claude Levi-Strass.(!) (In turn, my knowledge of it stems from a mention of it in the book "Algebra" by T.T. Moh.) I can't find a good summary online, so:
Anthropological Part
The following kinship system (the "Murngin system") is found apparently often in primitive societies:
(1) Every member of the society is assigned a type, with marriage only permissible for couples of the same type
(2) A person's type is a determined solely by their gender and the type of their parents; in the opposite direction, the type of a person's parents can be determined by their gender and type.
(3) Blood relations only determine whether or not two people have the same type. (So if there is one instance of a father and son having the same type, all fathers and sons have the same type -- if this is opaque, it will be rephrased more transparently using group theoretic language below.)
(4) The type of brothers and sisters is always different.
(5) It is always possible for some descendants of any two people to get married.
An anthropologist might observe that first cousins whose mothers are sisters never marry, or that when there less than 4 types of people in a tribe a marriage between first cousins whose parents are oppositely gendered siblings is permissible, and wonder whether these are separate empirical observations, or already implied by observations made above. Some elementary group theory gives the answer.
Group Theoretic Part
(1)&(2) rephrased: If we let the types of people be $1, 2, ..., n$, then by (2) we can define the type of the son of a couple of type $i$ to be $s(i)$, and the type of a daughter to be $d(i)$. The second part of (2) is just the statement that $s$ and $d$ are group actions on the set of types.
(3) rephrased: If $H$ is the subgroup of $S_n$ generated by $s$ and $d$, we can rewrite (3) as the statement that for any element $g$ of $H$, if $g(i) = i$ for some $i$, then $g(i) = i$ for all $i$ (in other words $g=e$).
(4) rephrased: $s(i) \neq d(i)$, or equivalently $d^{-1}s(i) \neq i$, for all $i$.
(5) rephrased: The orbit of any number $i$ under $H$ is the entire set of types $\{1,2,...,n\}$.
We can see right away then that if cousins share mothers who are sisters, they cannot marry, as if the type of the male is $i$, the type of the female is $ddd^{-1}s^{-1}(i) = ds^{-1}(i) \neq i$. This of course is not hard to see without group theory, but we can do more.
Plainly if the tribe is going to perpetuate itself there must be more than 1 type. If there are 2 types, then $s$ must be $(12)$ (it cannot be $e$), and likewise for $d$, a contradiction.
In the case of 3 types, then it is a matter of running through possibilities to show $s = d^{-1} = (123)$ up to permutation of type.
In the case of 4 types, a similar but more lengthy computation yields that up to permutation of type, there are the following four possibilities:
- $s = (1234)$, $d=e$
- $s = (1234)$, $d = (13)(24)$
- $s = (1234)$, $d = (1432)$
- $s = (12)(34)$, $d = (13)(24)$.
In both the case of 3 different types and 4 different types, it is now easy to verify that the group generated by $s$ and $d$ is commutative, proving that the statement about permissible marriages made earlier will always hold for such a type system.
...Hopefully at least a little interesting. I suppose some of the more geometric examples above might make for a more visceral application of group theory, at least on first exposure.