Central Type Group Algebra

• Central Type Group Algebra is defined by finite groups having a nondegenerate 2-cocycle, producing twisted algebras that are isomorphic to full matrix algebras.
• It interconnects group cohomology and projective representation theory, using character-theoretic criteria to relate element orders with irreducible representation dimensions.
• Its classification includes abelian cases, groups of order p^4, and square- or cube-free orders, highlighting explicit constructions and the structure of semi-centers.

A group of central type is a finite group $G$ for which there exists a nondegenerate $2$-cocycle $f \in Z^2(G, \mathbb{C}^\times)$. The associated twisted group algebra $\mathbb{C}^f G$ is then simple, i.e., isomorphic to a full matrix algebra. The study of central type groups and their twisted group algebras forms a central object in the intersection of group cohomology, projective representation theory, and the structure theory of semisimple algebras, with deep connections to gradings of matrix algebras and the intrinsic fundamental group of semisimple algebras (Schnabel, 2014, Ginosar et al., 2016).

1. Twisted Group Algebras and Nondegenerate 2-Cocycles

Given a finite group $G$ and a $2$-cocycle $f: G \times G \to \mathbb{C}^\times$ satisfying

$$f(g_{1},g_{2})\,f(g_{1}g_{2},g_{3}) = f(g_{2},g_{3})\,f(g_{1},g_{2}g_{3}),$$

the twisted group algebra $\mathbb{C}^f G$ has basis $\{u_g \mid g \in G\}$ and multiplication $u_x u_y = f(x, y)u_{xy}$, extended linearly. Associativity follows from the cocycle condition. By Maschke's theorem, $\mathbb{C}^f G$ is semisimple and decomposes as a sum of simple matrix algebras: $$\mathbb{C}^f G \cong \bigoplus_{i=1}^r M_{n_i}(\mathbb{C}).$$ A $2$-cocycle $f$ is called nondegenerate if the only $f$-regular element (i.e., $g$ such that $f(g, h) = f(h, g)$ for all $h \in C_G(g)$) is the identity. In this case, $\mathbb{C}^f G$ is itself simple and isomorphic to $M_n(\mathbb{C})$ for $n^2 = |G|$ (Schnabel, 2014).

2. Character-Theoretic and Cohomological Criteria

The central type condition is equivalent to the existence of an irreducible (linear) character $\chi$ of $G$ with $\chi(1)^2 = |G : Z(G)|$. This follows from the correspondence between projective representations associated to $f$ and irreducible modules of the twisted group algebra, as dictated by Schur's lemma. More cohomologically, $G$ is of central type if a nondegenerate class $[f] \in H^2(G, \mathbb{C}^\times)$ exists (Schnabel, 2014, Ginosar et al., 2016).

3. Classification of Groups of Central Type

The classification problem seeks to determine all finite groups (up to isomorphism) admitting a nondegenerate cocycle. Several structural results are known:

3.1 Abelian Groups

For any abelian $G$, central type is equivalent to $G \cong A \times A$ for some finite abelian group $A$, i.e., $G$ is a direct product of two identical abelian groups (Schnabel, 2014).

3.2 Groups of Order $p^4$

For $|G| = p^4$, where $p$ is prime, the central-type groups are classified as follows (Schnabel, 2014):

Prime $p$	Abelian central type	Nonabelian central type, representatives
Any $p$	$C_{p^2} \times C_{p^2}$, $C_p^4$
$p$ odd		(i) $$G_{(viii)} = \langle a, b \mid a^{p^2} = b^{p^2} = 1, [a, b] = a^p \rangle$$
		(ii) $$G_{(xiv)} = \langle a,b,c,d \mid a^p = b^p = c^p = d^p = 1, [c,d]=a, [a,-]=[b,-]=1\rangle$$
		(iii) $$G_{(xv)} = \langle a,b,c,d \mid a^p = b^p = c^p = d^p = 1, [d,b]=a, [d,c]=b, [a,-]=[b,-]=[c,-]=1\rangle$$
$p = 2$	$C_4 \times C_4$	$$\langle a, b, c \mid a^4 = b^2 = c^2 = 1, [a, b] = -1, [c, -] = 1 \rangle$$ (etc.)

3.3 Square-Free and Cube-Free Orders

For square-free $n$, there is a unique (abelian) group of central type $C_n \times C_n$ of order $n^2$ if and only if $p_j \not\equiv \pm 1 \pmod{p_i}$ for all prime divisors $p_i, p_j$. If $n$ is cube-free, all central-type groups are semidirect products of abelian central-type Sylow subgroups, with the semidirect product acting via a symplectic action, i.e., preserving a nondegenerate class in $H^2$ (Ginosar et al., 2016).

4. Semi-Center Structure and Commutativity

If $A$ is a $G$-module algebra (with $G$ acting by conjugation), the semi-center is defined as

$$\mathrm{Sz}(A) = \bigoplus_{\chi \in \widehat{G}} \{x \in A : g \cdot x = \chi(g)x\}.$$

For simple twisted group algebras $\mathbb{C}^f G$:

• If $G'$ (derived subgroup) is a Hall subgroup and $f|_{G'}$ remains nondegenerate, then $$\mathrm{Sz}(\mathbb{C}^f G) \cong \mathbb{C}^{f'} G'$$ is itself simple.
• The semi-center is commutative if and only if the natural map $\pi: \widehat{G} \rightarrow G/G'$ is zero, or equivalently, if $[f]$ is trivial/cohomologically trivial on every abelian subgroup (Schnabel, 2014).

The unique (for odd $p$) group $G_{(xv)}$ of order $p^4$ admits a nondegenerate cocycle with commutative semi-center; for $1 < |G| < 64$, the semi-center is noncommutative except in these sharp exceptions.

5. Explicit Constructions and Examples

Explicit nondegenerate cocycles in central type cases are constructed via crossed product techniques. For $G_{(viii)}$, one builds a crossed product $\mathbb{C}[a] \#_\zeta \mathbb{C}[b]$ using a $\mathbb{C}[a]$-automorphism: $b: a \mapsto \zeta_{p^2} a$ ($\zeta_{p^2}=e^{2\pi i/p^2}$). This yields twisted commutation relations with $u_b u_a = \zeta_{p^2} u_a u_b$, ensuring nondegeneracy of $f$ (Schnabel, 2014).

A nonabelian example with commutative semi-center is provided by $G_{(xv)}$ with a carefully constructed $f$ ensuring that semi-invariants arise only from the identity weight, rendering the semi-center commutative.

For small order groups, e.g., the dihedral group $D_4$ with a particular $2$-cocycle provides a noncommutative twisted group algebra whose semi-center is 4-dimensional and commutative (Schnabel, 2014).

6. Central Type, Gradings, and the Fundamental Group of Semisimple Algebras

Maximal connected gradings of $M_n(\mathbb{C})$ correspond bijectively to central type pairs $(G, y)$, where $G$ is of central type with $|G| \mid n^2$, and $y$ is an $\operatorname{Aut}(G)$-orbit of nondegenerate $H^2(G, \mathbb{C}^\times)$-classes. Every such grading is induced from a simple grading on $\mathbb{C}^f G$ by extending to the free product $\mathbb{F}_{d-1} * G$ (Ginosar et al., 2016).

The family of positive integers $n$ (denoted $\Lambda$) for which there is a unique group of central type of order $n^2$ is characterized as those for which $C_n \times C_n$ is the only central-type group, i.e., for all prime divisors $p_i, p_j$ of $n$, $p_j \not\equiv \pm 1 \pmod{p_i}$. This set consists of all square-free $n$ satisfying this arithmetic condition (Ginosar et al., 2016).

The intrinsic fundamental group $\pi_1(A)$ of a finite-dimensional semisimple algebra $A$ is the inverse limit of all connected group gradings under quotient gradings. For $A = M_n(\mathbb{C})$, this fundamental group is computed as an inverse limit over all $d \mid n$ with $d^2 = |G|$, using the free product $\mathbb{F}_{d-1} * (C_d \times C_d)$, reflecting a direct connection between gradings and central type groups (Ginosar et al., 2016).

7. Structural Properties, Extensions, and Closure Results

Central type is preserved under certain group extensions: if $N$ and $K$ are central type groups of coprime order, $G = N \rtimes K$ is of central type if and only if the induced action $K \to \operatorname{Aut}(N)$ is symplectic, i.e., it preserves a nondegenerate cohomology class in $H^2(N, \mathbb{C}^\times)$. This allows an inductive understanding of central type structures in terms of their Sylow subgroups and their interaction through automorphism groups (Ginosar et al., 2016).

The classification and construction of central type groups, their associated twisted group algebras, the behavior of semi-centers, and the connections to maximal connected gradings play an essential role in the broader framework of the representation theory and cohomology of finite groups, as well as the theory of graded and semisimple algebras (Schnabel, 2014, Ginosar et al., 2016).