Abstract algebra

Math 8 Homework 19 Spring, 2018

Due: Friday, May 25th at the start of class

1. In his 1878 paper1 that helped found group theory, Cayley claimed that there were three groups of order 6. The first was the cyclic group 〈 a 〉 = {e, a, a2, a3, a4, a5} (with a6 = e) and the other two were of the form {e, y, y2, x, xy, xy2} (with x2 = e = y3). These last two are distinguished by how x and y interact: in one we have xy = yx and the other we have xy = y2x and xy2 = yx. In what we call presentation notation, Cayley claimed three groups of order 6:

G1 = ⟨ a ∣∣ a6 = e⟩

G2 = ⟨ x, y

∣∣ x2 = e = y3, xy = yx⟩ G3 =

⟨ u, w

∣∣ u2 = e = w3, uw = w2u⟩ . I hope this is clear! For example, G2 consists of all products of x and y. For example xxxxyyxyxyxxxyyyy is an element. We might write this as x4y2xyxyx3y4. But we also know that x2 = e, so this simplifies to y2xyxyxy4. Similarly, y3 = e, so we could also write this as y2xyxyxy or y−1xyxyxy. We’re also told that xy = y2x, so we can “move” the xs to the right and the ys to the left:

y2xyxyxy = y2(y2x)(y2x)(y2x) = y4xy2xy2x.

Now it’s easiest to note that xy2 = (xy)y = y2xy = y4x = yx, so

y2xyxyxy = y4(xy2)(xy2)x = y4(yx)(yx)x = y5(xy)x2 = y5(y2x)x2 = y7x3 = yx.

The point is simply that while we are permitted any products of any number of xs and ys, we can always write it as ymxn (where m ∈{0, 1, 2} and n ∈{0, 1}). Now for the punchline (and the problem): There are really only two non-isomorphic groups of order 6. Please identify which of Professor Cayley’s groups are isomorphic, and prove to him that you are correct. (Here the simplest proof would be to explicitly write down an isomorphism between the two isomorphic groups.)

Hint: Can you find a structural property shared by two – and only two – of these three groups?

2. Product Groups. Let m and n be postive integers that are at least 2. Is Zm × Zn isomorphic to Zn × Zm? Remember we defined the product of two groups (G, · ) and (H, • ) as the set G × H with the operation induced from G and H: (g1, h1) ∗ (g2, h2) = (g1 ·g2, h1 •h2).

3. Let G be a finite group of order |G| = N.

(a) Prove that if g ∈ G has order k, then k divides N. Hint: Look at the subgroup H = 〈g 〉 = {e, g, g2, g3, . . .}. What does Lagrange’s Theorem say about |H|?

(b) Prove that for any g ∈ G we have gN = e.

1See http://www.jstor.org/stable/2369433.