This is a binary function whose first argument should be a group
G and whose second argument should be a natural number n.
It refers to the direct product of n copies of G.
This symbol represents a binary function with two arguments,
the first is a group G and the second a prime number p.
When applied to G and p, it represents a Sylow p-subgroup of G
(which is unique up to conjugacy in G).
The unary function whose value is the subgroup of argument
generated by all products of the form xyx^-1y^-1.
Commented Mathematical property (CMP):
d in the derived subgroup of G if and only if
there exist lists x,y of elements of G of equal length
such that d
is the product x_1 y_1 x_1^(-1) y_1^(-1) ... x_n y_n x_n^(-1) y_n^(-1).
This symbols represents a unary function whose argument should be a group G.
Its value is the biggest subgroup of G all of whose elements
commute with all elements of G.
Commented Mathematical property (CMP):
d is in the center of G if and only if
for all g in G we have g d= d g.
This symbols represents a binary function whose first argument should be a
group G and whose second argument should be an element g or a list of elements
L of the group G.
Its value is the subgroup of G of all elements
commuting with g or, if the second argument is a list, all elements of L.
Commented Mathematical property (CMP):
d is in the centralizer of g in G if and only if
g d= d g.
This symbol represents a unary function. The argument is a list or a
set. When evaluated on such an argument, the function represents the
free group generated by the entries of the list or set.
This symbol is a function with one argument, which should be a
vector space or a module V. When applied to
V it represents the group of all invertible linear transformations of V.
This symbol is a function with one argument, which should be a a
module V over a commutative ring. When applied to V it represents the
group of all invertible linear transformations of V of determinant 1.
This symbol is a function with two arguments. The first should be a positive
integer n, the second a
field F. When applied to
n and F it represents the group of all invertible linear transformations of
the vector space over F of dimension n.
This symbol is a function with two arguments. The first should
be a positive integer n, the second a field F. When applied to n and F it
represents the group of all invertible linear transformations of the vector
space over F of dimension n having determinant 1.
This symbols represents a binary function whose first argument should be a
group G and whose second argument should be a set of elements
or a subgroup L of the group G.
Its value is the subgroup of G of all elements
normalizing L.
Commented Mathematical property (CMP):
d is in the normalizer of X in G if and only if
g X= X g.
This symbol is a function with one argument, which should be
a natural number n. When applied to n
it represents the group of all permutations on the set {1,2,... ,n}.
Commented Mathematical property (CMP):
The carrier set of symmetric_groupn(k) consists of all permutations with
support in the integers {1,...,k}.
This symbol is a function with one argument, which should be
a natural number n. When applied to n
it represents the group of all even permutations on the set {1,2, ...,n}.