Theoria catervarum vel grupporum^[1][2] in mathematica et algebra abstracta structuras algebraicas cognoscit appellatas catervas. Notio catervae est media algebrae abstractae pars: aliae notissimae structurae algebraicae, sicut anelli (Anglice: rings), corpora, et spatia vectorum (Anglice: vector spaces) videri possunt catervae quae operationibus axiomatibusque additis praeditae sunt.

Catervae per omnem mathematicam fiunt, et rationes theoriae catervarum multas algebrae partes valide moverunt. Lineares catervae algebraicae et catervae Lie sunt rami theoriae catervarum qui progressum ingentem sustinuerunt ut res studii sui iuris fierent.

Varia systemata physica, sicut crystalli et atomus hydrogenii, a catervis symmetricis fingi possunt. Ergo theoriae catervarum et repraesentationis, arte conexae, multos usus in physica et chemia habent.

Una ex gravissimis confectionibus in mathematica saeculi vicensimi fuit opera communis, plus quam 10 000 paginas diurnariorum comprehendens, plerumque inter 1960 et 1980 editas, quae in classificatione finitarum catervarum simplicium fastigium habuit.

Caterva est copia G cum operatione °: G × G → G. G sub operatione clauditur: si a et b sunt elementa G, etiam a ° b est in G. Operatio est associativa: (a ° b) ° c = a ° (b ° c), semper. Non autem necesse est operationem commutativam esse: a ° b ≠ b ° a, generaliter. Si operatio commutativa est, caterva dicitur Abeliana.

Unum elementum e est idemfactor: e ° a = a, omnibus a in G.

Elementum quoddam a elementum inversum habet, ${\displaystyle a^{-1}}$, ut ${\displaystyle a^{-1}\circ a=e}$.

1. Idemfactor in dextera parte etiam elementum non mutat: a ° e = a. Demonstratio: pone ${\displaystyle a\circ e=x}$. Tunc ${\displaystyle a^{-1}\circ (a\circ e)=a^{-1}\circ x.}$ Sed ${\displaystyle a^{-1}\circ (a\circ e)=(a^{-1}\circ a)\circ e=e\circ e=e}$. Scimus ergo ${\displaystyle e=a^{-1}\circ x,}$, hoc est ${\displaystyle x=a}$, et ${\displaystyle a\circ e=a}$

2. Elementa inversa quoque in dextera parte aguntur: ${\displaystyle a\circ a^{-1}=e}$. Demonstratio: si ${\displaystyle a\circ a^{-1}=x}$, tunc ${\displaystyle (a^{-1}\circ a)\circ a^{-1}=a^{-1}\circ x}$. Hoc est, ${\displaystyle e\circ a^{-1}=a^{-1}\circ x}$. Quia ${\displaystyle e\circ a^{-1}=a^{-1}}$, scimus ${\displaystyle a^{-1}=a^{-1}\circ x}$, ergo ${\displaystyle x=e}$.

Subcaterva est subcopia elementorum G quae est caterva.

3. Subcopia H copiae G est caterva si:

• si ${\displaystyle h_{1},h_{2}\in H}$, tunc ${\displaystyle h_{1}\circ h_{2}\in H}$
• si ${\displaystyle h_{1}\in H}$, tunc ${\displaystyle h_{1}^{-1}\in H}$

Demonstratio: H sub operatione clauditur. e est elementum H quia ${\displaystyle h_{1}\circ h_{1}^{-1}=e}$ et scimus producta duorum elementorum H esse in H. Et inversa omnium H elementorum etiam in H sunt; H est igitur caterva.

• Augustinus Ludovicus Cauchy
• Evaristus Galois
• Iosephus Ludovicus Lagrange
• Felix Klein
• Theoria copiarum
• Vector (mathematica)

• Borel, Armand. 1991. Linear algebraic groups. Graduate Texts in Mathematics, 126. Ed. 2a. Berolini, Novi Eboraci: Springer-Verlag. MR1102012. ISBN 978-0-387-97370-8.
• Carter, Nathan C. 2009. Visual group theory. Classroom Resource Materials Series, Mathematical Association of America. MR2504193. ISBN 978-0-88385-757-1. http://web.bentley.edu/empl/c/ncarter/vgt/.
• Cannon, John J. 1969. "Computers in group theory: A survey", Communications of the Association for Computing Machinery 12: 3–12, doi:10.1145/362835.362837, MR0290613
• Frucht, R. 1939. "Herstellung von Graphen mit vorgegebener abstrakter Gruppe." Compositio Mathematica 6:239–250. ISSN 0010-437X, https://web.archive.org/web/20081201083831/http://www.numdam.org/numdam-bin/fitem?id=CM_1939__6__239_0.
• Golubitsky, Martin, et Ian Stewart. 2006 "Nonlinear dynamics of networks: the groupoid formalism." Bull. Amer. Math. Soc. (N.S.) 43:305–364. doi:10.1090/S0273-0979-06-01108-6, MR2223010.
• Judson, Thomas W. 1997. Abstract Algebra: Theory and Applications. http://abstract.ups.edu.
• Kleiner, Israel. 1986. "The evolution of group theory: a brief survey." Mathematics Magazine 59(4):195–215. MR863090. ISSN 0025-570X.
• La Harpe, Pierre de. 2000. Topics in geometric group theory. Sicagi: University of Chicago Press. ISBN 978-0-226-31721-2.
• Livio, M. 2005. The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry. Simon & Schuster. ISBN 0-7432-5820-7.
• Mumford, David. 1970 Abelian Varieties. Oxoniae: Oxford University Press. ISBN 978-0-19-560528-0. OCLC 138290.
• Ronan M. 2006. Symmetry and the Monster. Oxoniae: Oxford University Press. ISBN 0-19-280722-6.
• Rotman, Joseph. 1994. An Introduction to the Theory of Groups. Novi Eboraci: Springer-Verlag. ISBN 0-387-94285-8.
• Schupp, Paul E., et Roger C. Lyndon. 2001. Combinatorial Group Theory. Berolini, Novi Ebroaci: Springer-Verlag. ISBN 978-3-540-41158-1.
• Scott, W. R. (1964) 1987. Group Theory. Novi Eboraci: Dover. ISBN 0-486-65377-3.
• Shatz, Stephen S. 1972. Profinite Groups, Arithmetic, and Geometry. Princeton University Press. MR0347778. ISBN 978-0-691-08017-8.
• Weibel, Charles A. 1994. An Introduction to Homological Algebra. Cambridge Studies in Advanced Mathematics, 38. Cantabrigiae: Cambridge University Press. MR1269324. ISBN 978-0-521-55987-4, OCLC 36131259.