Primary Group (other)

	Primary Group (other) A primary group may refer to: * In mathematics, a special kind of group: a ''p''-primary group, also called simply ''p''-group; or a primary cyclic group In mathematics, a primary cyclic group is a group that is both a cyclic group and a ''p''-primary group for some prime number ''p''. That is, it is a cyclic group of order ''p'', C, for some prime number ''p'', and natural number ''m''. Every f ..., which is a ''p''-primary cyclic group. * In sociology, a primary group as opposed to secondary group. {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	P-primary Group In mathematics, specifically group theory, given a prime number ''p'', a ''p''-group is a group in which the order of every element is a power of ''p''. That is, for each element ''g'' of a ''p''-group ''G'', there exists a nonnegative integer ''n'' such that the product of ''pn'' copies of ''g'', and not fewer, is equal to the identity element. The orders of different elements may be different powers of ''p''. Abelian ''p''-groups are also called ''p''-primary or simply primary. A finite group is a ''p''-group if and only if its order (the number of its elements) is a power of ''p''. Given a finite group ''G'', the Sylow theorems guarantee the existence of a subgroup of ''G'' of order ''pn'' for every prime power ''pn'' that divides the order of ''G''. Every finite ''p''-group is nilpotent. The remainder of this article deals with finite ''p''-groups. For an example of an infinite abelian ''p''-group, see Prüfer group, and for an example of an infinite simple ''p''-g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Primary Cyclic Group In mathematics, a primary cyclic group is a group that is both a cyclic group and a ''p''-primary group for some prime number ''p''. That is, it is a cyclic group of order ''p'', C, for some prime number ''p'', and natural number ''m''. Every finite abelian group ''G'' may be written as a finite direct sum of primary cyclic groups, as stated in the fundamental theorem of finite abelian groups: :G=\bigoplus_\mathrm_ . This expression is essentially unique: there is a bijection between the sets of groups in two such expressions, which maps each group to one that is isomorphic. Primary cyclic groups are characterised among finitely generated abelian groups as the torsion groups that cannot be expressed as a direct sum of two non-trivial groups. As such they, along with the group of integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]