 Solution

ID: past-exam-of-the-lie-algebras-and-representation-theory-of-symmetric-groups-course-of-the-university-of-oxford/2008/a/1/d/solution

Let

M = L_{r}

and write

M_{k}

for its own lower central series. Part (c) proves inductively that

M_{k} \subseteq L_{r k}

: it holds at

k = 1

, and

[M, M_{k}] \subseteq [L_{r}, L_{r k}] \subseteq L_{r (k + 1)}

. If

L

has class less than

rs

, then

L_{rs} = 0

, so

M_{s} = 0

. Consequently

class (L_{r}) < s

, with the zero algebra understood to have class zero.

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