Maximal unipotent subgroup of symplectic group:Sp(4,p)

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Definition

Definition in terms of wreath product

Let p be a prime number greater than 2. This group G is a group of order p^4 defined as:

\! G := K/[[[K,K],K],K]

where K is the wreath product of groups of order p and [[[K,K],K],K] is the fourth member of the lower central series of K.

Definition in terms of symplectic group

Let p be a prime number greater than 2. This group is defined as the p-Sylow subgroup of the symplectic group of degree four over the prime field \mathbb{F}_p. Another way of saying this is that it is a maximal unipotent subgroup of symplectic group of degree four for a field of prime size p.

Particular cases

This group definition works only for p \ge 3:

Value for prime number p Corresponding group
3 wreath product of Z3 and Z3
5 quotient of wreath product of Z5 and Z5 by fourth member of lower central series

GAP implementation

Group ID

This finite group has order p^4 and has ID 7 among the group of order p^4 in GAP's SmallGroup library. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(p^4,7)

For instance, we can use the following assignment in GAP to create the group and name it G:

gap> G := SmallGroup(p^4,7);

Conversely, to check whether a given group G is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [p^4,7]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.

Other descriptions

This group can be described using GAP as:

gap> K := WreathProduct(CyclicGroup(p),CyclicGroup(p));
<group of size 15625 with 2 generators>
gap> G := K/CommutatorSubgroup(CommutatorSubgroup(DerivedSubgroup(K),K),K);