Group cohomology of elementary abelian group:E16

This article gives specific information, namely, group cohomology, about a particular group, namely: elementary abelian group:E16.
View group cohomology of particular groups | View other specific information about elementary abelian group:E16

Family contexts

Family	Parameter values	Information on group cohomology of family
elementary abelian group of order $p^{n}$ for a prime $p$	$p=2,n=4$	group cohomology of elementary abelian groups
elementary abelian group of prime-fourth order for a prime $p$	$p=2$	group cohomology of elementary abelian group of prime-fourth order

Homology groups for trivial group action

FACTS TO CHECK AGAINST (homology group for trivial group action):
First homology group: first homology group for trivial group action equals tensor product with abelianization
Second homology group: formula for second homology group for trivial group action in terms of Schur multiplier and abelianization|Hopf's formula for Schur multiplier
General: universal coefficients theorem for group homology|homology group for trivial group action commutes with direct product in second coordinate|Kunneth formula for group homology

Over the integers=

The homology groups over the integers are given as follows:

$\!H_{q}(E_{16};\mathbb {Z} )=\left\lbrace {\begin{array}{rl}(\mathbb {Z} /2\mathbb {Z} )^{\frac {2q^{3}+15q^{2}+34q+45}{24}},&\qquad q=1,3,5,\dots \\(\mathbb {Z} /2\mathbb {Z} )^{\frac {2q^{3}+15q^{2}+34q}{24}},&\qquad q=2,4,6,\dots \\\mathbb {Z} ,&\qquad q=0\\\end{array}}\right.$

The first few homology groups are given as follows:

$q$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$
$H_{q}$	$\mathbb {Z}$	$(\mathbb {Z} /2\mathbb {Z} )^{4}$	$(\mathbb {Z} /2\mathbb {Z} )^{6}$	$(\mathbb {Z} /2\mathbb {Z} )^{14}$	$(\mathbb {Z} /2\mathbb {Z} )^{21}$	$(\mathbb {Z} /2\mathbb {Z} )^{35}$	$(\mathbb {Z} /2\mathbb {Z} )^{49}$	$(\mathbb {Z} /2\mathbb {Z} )^{71}$	$(\mathbb {Z} /2\mathbb {Z} )^{94}$
rank of $H_{q}$ as an elementary abelian 2-group	--	4	6	14	21	35	49	71	94