COHOMOLOGY OF GROUPS : EXERCISES

Sheet 6, 3 July 2007

Exercise 6.1. Determine the edge homomorphisms for the Lyndon-Hochschild-Serre spectral sequence in cohomology.

Exercise 6.2. Compute the integral cohomology ring H^{∗}(D_{2m}; ℤ) of the dihedral group D_{2m} with m odd, using an extension

1 → C_{m} → D_{2m} → C_{2} → 1.

Exercise 6.3. Let S_{k} denote the symmetric group with k! elements. Construct a suitable inclusion S_{m} ≀ S_{n} → S_{mn}, and use it to show that the wreath product is associative.

Exercise 6.4. Show that C_{p} ≀ ⋯ ≀ C_{p}, with r copies of C_{p}, embeds as a p-Sylow subgroup in S_{p^{r}}.

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