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G is finite where the number of subgroups in g is finite?
Actually a stronger statement can be made:A group G is finite if and only if the number of its subgroups is finiteLet G be a group. If G is finite there is only a finite number of subsets of G, so clearlya finite number of subgroups.Now suppose G is infinite , let'ssuppose one element has infinite order. The this element generates an infinite cyclicgroup which in turn contains infinitely many subgroups.Now suppose all the subgroups have finite order Take some element of G and let it generate a finite group H. Now take another element of G not in H and let it generate a finite group I. Keep doing this by next picking an element of G not H or I. You can continue this way.
Is turquoise apart of a mineral group or mineral class?
Turquoise is a member of the turquoise group and is classed as a phosphate. Phosphates are a class of minerals that is part of a large and diverse group of minerals.
What is a group of students called collective noun?
no. it must be class
What is the syllabus for play group?
You don't need a syllabus for a play group. It is play and not a class. A syllabus is used for a college class to tell the students what chapters they need to read by certain dates and when the tests are.
Which new social group emerged in 18th century in france?
middle class
G is finite group if and only if the number of it is subgroups is finite?
False. G may be a finite group without sub-groups.
What is a Definition for algebraic equation?
an equation in the form of a polynomial having a finite number of terms and equated to zeroan equation in the form of a polynomial having a finite number of terms and equated to zero
Every finite group is isomorphic to a permutation group?
yes form cayleys theorem . every group is isomorphic to groups of permutation and finite groups are not an exception.
What is an alternating group?
In group theory, an alternating group is a group of even permutations of a finite set.
What has the author Wolfgang Hamernik written?
Wolfgang Hamernik has written: 'Group algebras of finite groups' -- subject(s): Finite groups, Group algebras
What has the author Michael Aschbacher written?
Michael Aschbacher has written: '3-transposition groups' -- subject(s): Finite groups 'The classification of finite simple groups' -- subject(s): Group theory and generalizations -- Abstract finite groups -- Finite simple groups and their classification, Finite simple groups, Representations of groups, Group theory and generalizations -- Representation theory of groups -- Modular representations and characters 'Fusion systems in algebra and topology' -- subject(s): Combinatorial group theory, Topological groups, Algebraic topology 'The classification of quasithin groups' -- subject(s): Classification, Finite simple groups 'Finite group theory' -- subject(s): Finite groups
G is finite where the number of subgroups in g is finite?
Actually a stronger statement can be made:A group G is finite if and only if the number of its subgroups is finiteLet G be a group. If G is finite there is only a finite number of subsets of G, so clearlya finite number of subgroups.Now suppose G is infinite , let'ssuppose one element has infinite order. The this element generates an infinite cyclicgroup which in turn contains infinitely many subgroups.Now suppose all the subgroups have finite order Take some element of G and let it generate a finite group H. Now take another element of G not in H and let it generate a finite group I. Keep doing this by next picking an element of G not H or I. You can continue this way.
What is an affine group?
An affine group is the group of all affine transformations of a finite-dimensional vector space.
Is every finite abelian group is cyclic?
No, for instance the Klein group is finite and abelian but not cyclic. Even more groups can be found having this chariacteristic for instance Z9 x Z9 is abelian but not cyclic
What is finite and infinite cyclic group?
Normally, a cyclic group is defined as a set of numbers generated by repeated use of an operator on a single element which is called the generator and is denoted by g.If the operation is multiplicative then the elements are g0, g1, g2, ...Such a group may be finite or infinite. If for some integer k, gk = g0 then the cyclic group is finite, of order k. If there is no such k, then it is infinite - and is isomorphic to Z(integers) with the operation being addition.
How do you prove that order of a group G is finite only if G is finite and vice versa?
(1). G is is finite implies o(G) is finite.Let G be a finite group of order n and let e be the identity element in G. Then the elements of G may be written as e, g1, g2, ... gn-1. We prove that the order of each element is finite, thereby proving that G is finite implies that each element in G has finite order. Let gkbe an element in G which does not have a finite order. Since (gk)r is in G for each value of r = 0, 1, 2, ... then we conclude that we may find p, q positive integers such that (gk)p = (gk)q . Without loss of generality we may assume that p> q. Hence(gk)p-q = e. Thus p - q is the order of gk in G and is finite.(2). o(G) is finite implies G is finite.This follows from the definition of order of a group, that is, the order of a group is the number of members which the underlying set contains. In defining the order we are hence assuming that G is finite. Otherwise we cannot speak about quantity.Hope that this helps.
What are permutation groups?
A permutation group is a group of permutations, or bijections (one-to-one, onto functions) between a finite set and itself.
