Author(s):

Jeff Suzuki (Brooklyn College)

**NOTE**: In the following, we'll gradually scale our way up towards the full Enigma encryption on 26 letters.

- Suppose \(E_{1}, E_{2} \) are two Enigma permutations, and \(E_{1} \) contains \((\alpha c) \) while \(E_{2} \) contains \((\alpha f) \). You should assume both contain a number of other transpositions as well.
- Prove: \(E_{2}E_{1} ( c) = f \). (In particular, why can we disregard the effect of the other transpositions in \(E_{2}, E_{1} \)?)
- Suppose \(E_{1}, E_{2} \) contained the same transposition \((\alpha\beta) \). Find \(E_{2}E_{1}(\alpha) \) and \(E_{2}E_{1}(\beta) \).

- Suppose \(\sigma = (ab)(cd)(ef) \) and \(\tau = (ab)(cf)(de) \) are two permutations on six symbols.
- Find the product \(\sigma\tau \).
- Classify the cycles in the product \(\sigma\tau \). In particular: How many 1-cycles? How many 2-cycles? How many 3-cycles?
- \((ab) \) is in both \(\sigma \) and \(\tau \). What do you notice about \(a, b \) in the product \(\sigma\tau \)?
- The transposition \((cd) \) is in \(\sigma \). What do you notice about \(c, d \) in the product \(\sigma\tau \)?
- The transposition \((de) \) is in \(\tau \). What do you notice about \(d, e \) in the product \(\sigma\tau \)?

- Suppose \(\alpha = (ab)(cd)(ef)(gh) \) and \(\beta = (ad)(bg)(ce)(fh) \) are two permutations on eight symbols.
- Find the product \(\alpha\beta \).
- Classify the cycles in the product \(\alpha\beta \). In particular: How many 1-cycles? How many 2-cycles? How many 3-cycles?
- What do you notice about the elements of each transposition of \(\alpha \)?

- Suppose \(\mu = (ab)(ce)(dj)(fg)(kl)(hi) \) and \(\nu = (aj)(cd)(db)(fk)(gl)(hi) \) are two permutations on twelve symbols.
- Find \(\mu\nu \).
- Classify the cycles in the product \(\mu\nu \).
- Suppose \((xy) \) is a transposition in exactly one of \(\mu \) or \(\nu \). What can you say about \(x, y \) in the product \(\mu\nu \)?

- Note that all the preceding involutions are proper. Suppose \(\kappa = (ab)(cd)(ef) \) and \(\lambda = (ac)(be) \) are two permutations on six symbols.
- Find \(\kappa\lambda \).
- What happens with products of proper involutions that does not happen when one of the involutions is not proper?

*Return to the overview of Part 3.1 (Cycle Decomposition).*

*Continue to the overview of Part 3.2 (Rejewski's Theorems).*

Jeff Suzuki (Brooklyn College), "The Theorem that Won the War: Activities for Part 3.1 (Cycle Decomposition)," *Convergence* (October 2023)