**Group Theory**

In 1973, Saharon Shelah showed that the Whitehead problem ("is every abelian group *A* with Ext1(A, **Z**) = 0 a free abelian group?") is independent of ZFC. A group with Ext1(A, **Z**) = 0 which is not free abelian is called a Whitehead group; MA + ¬CH proves the existence of a Whitehead group, while *V* = *L* proves that no Whitehead group exists.

Read more about this topic: List Of Statements Undecidable In ZFC

### Other articles related to "group theory, groups, theory, group":

**Group Theory**

... Applications of

**group theory**abound ... all structures in abstract algebra are special cases of

**groups**... Rings, for example, can be viewed as abelian

**groups**(corresponding to addition) together with a second operation (corresponding to multiplication) ...

... Frank Adams was a leading figure in algebraic topology and homotopy

**theory**... methods which led to important advances in calculating the homotopy

**groups**of spheres (a problem which is still unsolved), including the invention of the ... He is also known for his work in the

**theory**of statistical inference and in multivariate analysis ...

**Group Theory**- Late 20th Century

... years of the 20th century enjoyed the successes of over one hundred years of study in

**group theory**... In finite

**groups**, post classification results included the O'Nan–Scott theorem, the Aschbacher classification, the classification of multiply transitive finite

**groups**, the determination ... The modular representation

**theory**entered a new era as the techniques of the classification were axiomatized, including fusion systems, Puig's

**theory**of pairs and ...

**Group Theory**- Early 19th Century - Convergence

...

**Group theory**as an increasingly independent subject was popularized by Serret, who devoted section IV of his algebra to the

**theory**by Camille Jordan, whose Traité des substitutions et des équations ... Other

**group**theorists of the 19th century were Bertrand, Charles Hermite, Frobenius, Leopold Kronecker, and Émile Mathieu as well as Burnside, Dickson, Hölder, Moore, Sylow, and ... the above three sources into a uniform

**theory**started with Jordan's Traité and von Dyck (1882) who first defined a

**group**in the full modern sense ...

**Group Theory**- Considered Problems

... One problem considered in the study of combinatorics on words in

**group theory**is the following for two elements x,y of a semigroup, does x=y modulo the defining relations of x and y ... Undecidable means the

**theory**cannot be proved ... This question asks if a

**group**is finite if the

**group**has a definite number of generators and meets the criteria xn=1, for x in the

**group**...

### Famous quotes containing the words theory and/or group:

“Psychotherapy—The *theory* that the patient will probably get well anyway, and is certainly a damned ijjit.”

—H.L. (Henry Lewis)

“Unless a *group* of workers know their work is under surveillance, that they are being rated as fairly as human beings, with the fallibility that goes with human judgment, can rate them, and that at least an attempt is made to measure their worth to an organization in relative terms, they are likely to sink back on length of service as the sole reason for retention and promotion.”

—Mary Barnett Gilson (1877–?)