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Lagrange's Theorem
By damian Haigh (T256) on Sunday, July 4, 1999 - 05:31 pm:

I'm working through SMP 16-19 Mathematical Structure, looking at groups. I feel the need to ask some stupid questions - I hope you'll humour me.
The tasksheet on Lagrange's theorem asks me to look at the subgroups of S4 (meaning the group of permutations of 4 objects - can I call this a permutation group or something?). Anyway it makes sense to me that the group of transformations of a square (the symmetry group of a square?) is a subgroup of S4 which appears to be cyclic - is this correct?

Now I'm trying to get my head round Lagrange's theorem and I'm looking at the group S3 of transformations of an equilateral triangle. The argument seems to be that any subgroup will display a cyclic pattern eg

abcd
bcda
cdab
dabc

In which case their group table must be square, and it follows that squares can only be arranged within a larger square where the subgroup's order is a factor of the order of the larger group.

My initial questions are:
do all subgroups (not just those of the tranformations of regular polyhedra or any other cylic group) display the cyclic property? Therefore is any subgroup of a permutation group cylic?
If so how come the klein group is not cylic, since it must be a subgroup of a permutation group (presumably S4)?
Also: Why is it neccesary that the subgroups display this cyclic pattern; isn't it enough that their group tables must be square?

I know I'm missing something straightforward here but I'm short of time (big pile of marking to do after the Nrich conference!) so I'd appreciate it if you could point me in the right direction.
Damian

By damian Haigh (T256) on Wednesday, July 7, 1999 - 03:33 pm:

I now realise that only subgroups generated by a single element are cyclic, so it makes sense that the K group isn't cyclic.
I think I've got this now ... but what would you recommend for reference reading on this topic?

By Anonymous on Wednesday, August 4, 1999 - 03:52 pm:

Reply from Dr A.F.Beardon, 4 August 1999

Dear Damian,
I hope the following comments on Langrange's Theorem help.

(1) Yes, S4 is a permutation group. In general, a permutation of a set X is a one-to-one (ie injective) map of X onto itself, and the set of all such permutations form a group, called the permutation group of X. The permutation group of {1,2,...,n} is denoted (universally) by Sn.

(2) The group G of transformations (= rigid otions)of a square is a subgroup of S4 because each transformation of the square permutes the four vertices (ie is a one-to-one map of the set of four vertices onto itself). However, G is not all of S4 because we cannot keep two adjacent vertices fixed and interchange the other two by a rigid motion of the square. You rightly say that the group of rotations of the square (about a vertical axis when the square is lying in the horizontal plane) is a cyclic group of order 4. However, G also contains four other rotations (by an angle 180 degrees) whose axes lie along the four lines of symmetry IN the square. It follows that G has eight elements. In conclusion, G ontains a cyclic subgroup of rotations of order 4; G itself is of order 8, and it is a subgroup of S4 of order 24.

(3) Yes, a cyclic group is (by definition) a group that is generated by a single element. If a cyclic group has m elements, it must contain an element g such that e (the identity),g,g2,...,gm-1 are distinct, but that gm=e. As every element x of the Klein group satisfies x2=e, and as the Klein group has order 4, the Klein group is not cyclic.

(4) Suggested references.
There are many many good books on group theory all of which contain sections on permutation groups. The only books that I know of that deal with permutation groups ONLY are very advanced.
I mention four books that I think might be helpful.

(a) Budden, F.J.,
The Fascination of Groups,
Cambridge Univ. Press, 1972
[Nearly 600 pages; elementary, with very many examples]

(b) Ledermann, W.,
Introduction to Group Theory
Oliver and Boyd, 1973
[Short, a little terse, but very good]

(c) Burn, R.P.,
Groups : a Path to Geometry
Cambridge Univ. Press, 1985
[This consists of structured exercises only; the
theory is gradually developed through exercises.]

(d) Jordan, C.R. and Jordan, D.A.,
Groups,
Edward Arnold, 1994.
[I suggest this might be the most useful- it should
also be easily available, and is a paperback of
about 200 pages costing about 9 pounds.]
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