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A cyclic group is a group which is equal to one of its cyclic subgroups: G = g for some element g, called a generator of G. For a finite cyclic group G of order n we have G = {e, g, g2, ... , gn−1}, where e is the identity element and gi = gj whenever i ≡ j (mod n); in particular gn = g0 = e, and g−1 = gn−1.
This group has exponent p 2. If n is a positive integer there are two extraspecial groups of order p 1+2n, which for p odd are given by The central product of n extraspecial groups of order p 3, all of exponent p. This extraspecial group also has exponent p. The central product of n extraspecial groups of order p 3, at least one of exponent p 2.
For example, 3 5 = 3 · 3 · 3 · 3 · 3 = 243. The base 3 appears 5 times in the multiplication, because the exponent is 5. Here, 243 is the 5th power of 3, or 3 raised to the 5th power. The word "raised" is usually omitted, and sometimes "power" as well, so 3 5 can be simply read "3 to the 5th", or "3 to
For example, the cyclic group C 4 and the Klein four-group V 4 which is C 2 × C 2 are both 2-groups of order 4. There are three abelian groups of order p 3, namely C p 3, C p 2 × C p, and C p × C p × C p. There are also two non-abelian groups. For p ≠ 2, one is a semi-direct product of C p × C p with C p, and the other is a semi-direct ...
In many popular fonts the Unicode "superscript" and "subscript" characters are actually numerator and denominator glyphs. Unicode has subscripted and superscripted versions of a number of characters including a full set of Arabic numerals. [1] These characters allow any polynomial, chemical and certain other equations to be represented in plain ...
Abelian group. In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian ...
Order and structure. The order of a group G and the orders of its elements give much information about the structure of the group. Roughly speaking, the more complicated the factorization of | G |, the more complicated the structure of G. For | G | = 1, the group is trivial. In any group, only the identity element a = e has ord (a) = 1.
2. Denotes the additive inverse and is read as minus, the negative of, or the opposite of; for example, –2. 3. Also used in place of \ for denoting the set-theoretic complement; see \ in § Set theory. × (multiplication sign) 1. In elementary arithmetic, denotes multiplication, and is read as times; for example, 3 × 2. 2.
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