### Citations

234 |
On the structure of abstract algebras
- Birkhoff
- 1935
(Show Context)
Citation Context ...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... |

60 | A Course - Burris, Sankappanavar - 1981 |

23 |
Varieties of groups, Springer-Verlag
- Neumann
- 1967
(Show Context)
Citation Context ...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... |

18 |
The Kourovka notebook, Unsolved problems in group theory 16 (Russian Academy of Sciences
- Mazurov, Khukhro
- 2006
(Show Context)
Citation Context ...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. ... |

16 |
Zur Theorie der endlichen auflösbaren Gruppen
- Gaschütz
(Show Context)
Citation Context ...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... |

4 |
On Birkhoff’s theorem
- Kogalovskĭı
- 1965
(Show Context)
Citation Context ...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 ... |

1 |
Benthem, A note on Jónsson’s theorem, Algebra Universalis 25
- van
- 1988
(Show Context)
Citation Context ...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... |