# Notation for subgroup-defining functions

This is a notation page: it lists some important notation

For convenience of notation, $G$ here denotes the group on which we are computing the subgroup-defining function.

## Subgroup-defining functions

Name of subgroup-defining function Notation for single iteration Notation for $n$ iterations Associated quotient-defining function Notation for single iteration Notation for multiple iterations
Center $Z(G)$ -- (idempotent) Inner automorphism group $Inn(G), G/Z(G)$  ?
Commutator subgroup, derived subgroup $G'$, $[G,G]$ $G^{''''\ldots} n$ times, or $G^{(n)}$ Abelianization $G^{ab}$ -- (idempotent)
Fitting subgroup $F(G)$ -- (idempotent) Fitting quotient $G/F(G)$ -- (idempotent)
Frattini subgroup $\Phi(G)$ $\Phi^n(G)$ Frattini quotient $G/\Phi(G)$ -- (idempotent)
• $Soc(G)$ denotes the socle of $G$
• $W(G)$ denotes the Wielandt subgroup of $G$
• $O(G)$ denotes the Brauer core of $G$
• $E(G)$ denotes the layer of $G$
• $F^*(G)$ denotes the generalized Fitting subgroup of $G$
• $L(G)$ denotes the 2-layer of $G$