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...orphic images, subalgebras, finite products, and V is axiomatizable. (5) V is an axiomatic formation that is closed under subalgebras. Proof. The equivalence of (1) and (2) is Birkhoff’s Theorem (see =-=[2]-=-). Note that these equivalent conditions easily imply each of (3)–(5). The implication (3)⇒(1) is Kogalovskĭı’s Theorem (see [5]). The implication (4)⇒(1) can be deduced from the theorem of van Benth...

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...nt groups of exponent p, where p is an odd prime, is a nonabelian variety whose axiomatic subformations are subvarieties. Before starting, we identify the subvarieties of N p2 . By Corollary 35.12 of =-=[7]-=-, any k-step nilpotent variety V is generated by its free group on k generators, which we denote by FV(k). Therefore a subvariety V ⊆ N p2 is generated by FV(2), which is a homomorphic image of F = FN...

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...o refer to any class of groups that is closed under homomorphic images and finite subdirect products. Using this definition, A. Gaglione and D. Spellman ask in Problem 14.32 of The Kourovka Notebook, =-=[6]-=-, whether every first-order axiomatizable formation of groups is a variety, i.e., an equationally axiomatizable class. They mention that the answer is affirmative for any formation of abelian groups. ...

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...e group contains an axiomatic formation that is not a subvariety. 1. Introduction A formation of groups is a class of finite groups closed under homomorphic images and finite subdirect products. (See =-=[4]-=-.) The concept of a formation makes sense even if the groups are infinite, and in this paper we will use the word “formation” to refer to any class of groups that is closed under homomorphic images an...

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...bras. Proof. The equivalence of (1) and (2) is Birkhoff’s Theorem (see [2]). Note that these equivalent conditions easily imply each of (3)–(5). The implication (3)⇒(1) is Kogalovskĭı’s Theorem (see =-=[5]-=-). The implication (4)⇒(1) can be deduced from the theorem of van Benthem in [1] which states that an axiomatic class of algebraic structures is closed under homomorphic images and subalgebras if and ...

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...e that these equivalent conditions easily imply each of (3)–(5). The implication (3)⇒(1) is Kogalovskĭı’s Theorem (see [5]). The implication (4)⇒(1) can be deduced from the theorem of van Benthem in =-=[1]-=- which states that an axiomatic class of algebraic structures is closed under homomorphic images and subalgebras if and only if it is axiomatizable by sentences that are finite disjunctions of equatio...