Pentactions and action representability in the category of reduced groups with action

Abstract A notion of pentaction of any object in the category 𝐫𝐆𝐫 ∙ rGr^{} of reduced groups with action is introduced. The operations are defined in the set 𝖯𝖾𝗇𝗍𝖺𝖼𝗍 ⁢ (A) Pentact (A) of pentactions of an object A of 𝐫𝐆𝐫 ∙ rGr^{}. It is proved that if an object A is perfect with zero weak stabilizer in the sense defined in this paper, then 𝖯𝖾𝗇𝗍𝖺𝖼𝗍 ⁢ (A) Pentact (A) is an object of 𝐫𝐆𝐫 ∙ rGr^{}, it has a derived action on A, the object A is action representable and 𝖯𝖾𝗇𝗍𝖺𝖼𝗍 ⁢ (A) Pentact (A) represents all actions on A.