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 $G$