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COHOMOLOGY OF GROUPS : EXERCISES Sheet 7, 9 July 2007 Exercise 7.1. Let p > 3 be a prime, let C

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COHOMOLOGY OF GROUPS : EXERCISES Sheet 7, 9 July 2007

Exercise 7.1. Let p > 3 be a prime, let C

p

= ht|t

p

= 1i and C

p2

= hs|s

p2

= 1i be cyclic groups of order p and p

2

. We let C

p

act on C

p2

by t(s) = s

1+p

. Let E = C

p2

o C

p

be the corresponding extension

1 → C

p2

→ E → C

p

→ 1.

Compute H

∗

(E ; F

p

) as an algebra.

Hint: Consider the LHS spectral sequence associated to this extension. Show that it has a non-trivial d

2

differential by computing H

2

(E; F

p

) and comparing it with E

20,2

⊕ E

21,1

⊕ E

22,0

. To compute H

2

(E; F

p

), you can for example use an explicit resolution, or first compute H

i

(E ; Z ) for i = 2, 3 with the LHS spectral sequence and apply universal coefficients. Show that it collapses at the E

3

-term. Show that there are no multiplicative extensions.

Exercise 7.2. Compute H

∗

(D

8

; F

2

) as an F

2

-algebra.

Hint: Prove that there is an isomorphism D

8

∼ = C

2

o C

2

.

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