Heisenberg groups, semifields, and translation planes

Semi-extraspecial Groups

“…The following theorem has been proved in a number of places. See Lemma 3 in [4], Lemma 2.1 of [20], Proposition 2.2 of [27], the Theorem in [45], and page 139 of [51] when p is odd. F is a (pre-)semifield, then G(F ) is an ultraspecial group of order |F | 3 with at least two abelian subgroups of order |F | 2 .…”

“…It is not difficult to see that A 1 = {(a, 0, c) | a, c ∈ F } and A 2 = {(0, b, c) | b, c ∈ F } are abelian subgroups of order |F | 2 . If p is odd, G(F ) has exponent p (see Proposition 2.3 (5) of [27] or page 139 of [51]). For p = 2, the elements in A 1 ∪ A 2 have order 2 and every element of G(F ) outside of A 1 ∪ A 2 has order 4 (see Proposition 2.3 (6) of [27]).…”

“…If p is odd, G(F ) has exponent p (see Proposition 2.3 (5) of [27] or page 139 of [51]). For p = 2, the elements in A 1 ∪ A 2 have order 2 and every element of G(F ) outside of A 1 ∪ A 2 has order 4 (see Proposition 2.3 (6) of [27]). The following theorem by Verardi (Proposition 3.1 of [51]) yields the connection between semifields and ultraspecial groups with two abelian subgroups of maximal possible order.…”

“…It is possible to have F isotopic to F op when F is not isotopic to a commutative semifield. Both Hiranime (in Theorem 5.1 of [20]) and Knarr and Stroppel (Proposition 3.2 (b) of [27]) have proved:…”

“…We will see that the construction of these semifield groups has appeared in the finite geometry literature a number of times (see [10], [20], [27], and [45]). Our second goal is to present the results from the finite geometry literature regarding these groups.…”