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In Note M, Burnside (1955) states, "The contrast that these results shew between groups of odd and of even order suggests inevitably that simple groups of odd order do not ...

Let g be a finite-dimensional Lie algebra over some field k. A subalgebra h of g is called a Cartan subalgebra if it is nilpotent and equal to its normalizer, which is the ...

Let K be a number field, then each fractional ideal I of K belongs to an equivalence class [I] consisting of all fractional ideals J satisfying I=alphaJ for some nonzero ...

The categorical notion which is dual to product. The coproduct of a family {X_i}_(i in I) of objects of a category is an object C=coproduct_(i in I)X_i, together with a ...

The cross number of a zero-system sigma={g_1,g_2,...,g_n} of G is defined as K(sigma)=sum_(i=1)^n1/(|g_i|) The cross number of a group G has two different definitions. 1. ...

A cyclotomic field Q(zeta) is obtained by adjoining a primitive root of unity zeta, say zeta^n=1, to the rational numbers Q. Since zeta is primitive, zeta^k is also an nth ...

A module over a unit ring R is called divisible if, for all r in R which are not zero divisors, every element m of M can be "divided" by r, in the sense that there is an ...

A fractional ideal is a generalization of an ideal in a ring R. Instead, a fractional ideal is contained in the number field F, but has the property that there is an element ...

When referring to a planar object, "free" means that the object is regarded as capable of being picked up out of the plane and flipped over. As a result, mirror images are ...

A group is called a free group if no relation exists between its group generators other than the relationship between an element and its inverse required as one of the ...