Combinatorics Through Guided Discovery, 2004a
Combinatorics Through Guided Discovery, 2004a
Combinatorics Through Guided Discovery, 2004a
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6.1. Permutation Groups 111<br />
line) are different. Will this be true in any group table for a permutation<br />
group? Why or why not?<br />
Problem 267. In Table 6.1.4, every element of the group appears in every<br />
row (even if you don’t include the first element, the one before the line).<br />
Will this be true in any group table for a permutation group? Why or why<br />
not? Also in Table 6.1.4 every element of the group appears in every column<br />
(even if you don’t include the first entry, the one before the line). Will this<br />
be true in any group table for a permutation group? Why or why not?<br />
• Problem 268. Write down the group table for the dihedral group D 4 . Use<br />
the ϕ notation described above to denote the flips. (Hints: Part of the table<br />
has already been written down. Will you need to think hard to write down<br />
the last row? Will you need to think hard to write down the last column?<br />
When you multiply a product like ϕ 1|3 ◦ ρ remember that we defined ϕ 1|3<br />
to be the flip that fixes the vertex in position 1 and the vertex in position 3,<br />
not the one that fixes the vertex on the square labelled 1 and the vertex on<br />
the square labelled 3.)<br />
You may notice that the associative law, the identity property, and the inverse<br />
property are three of the most important rules that we use in regrouping parentheses<br />
in algebraic expressions when solving equations. There is one property we<br />
have not yet mentioned, the commutative law which would say that σ ◦ ϕ = ϕ ◦ σ.<br />
It is easy to see from the group table of R 4 that it satisfies the commutative law.<br />
Problem 269. Does the commutative law hold in all permutation groups?<br />
6.1.6 Subgroups<br />
We have seen that the dihedral group D 4 contains a copy of the group of rotations<br />
of the square. When one group G of permutations of a set S is a subset of another<br />
group G ′ of permutations of S, we say that G is a subgroup of G ′ .<br />
• Problem 270. Find all subgroups of the group D 4 . (h)<br />
Problem 271. Can you find subgroups of the symmetric group S 4 with two<br />
elements? Three elements? Four elements? Six elements? (For each positive<br />
answer, describe a subgroup. For each negative answer, explain why not.)
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