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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.
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Let G be a finite cyclic group prove that every subgroup H of G is isomorphic to some factor group G mod K and every factor group G mod K is isomorphic to some subgroup H?
Let ( G ) be a finite cyclic group generated by an element ( g ). Any subgroup ( H ) of ( G ) can be expressed as ( H = \langle g^k \rangle ) for some divisor ( k ) of the order of ( G ). The factor group ( G/K ) for some subgroup ( K ) is also cyclic, and by choosing ( K ) appropriately (for instance, ( K = \langle g^m \rangle ) where ( m ) divides the order of ( G )), we can ensure that ( G/K ) is isomorphic to ( H ). Thus, every subgroup ( H ) corresponds to a factor group ( G/K ) and vice versa, establishing the desired isomorphism relationships.
Data in order from smallest to largest or vice-versa?
numerical order
Data in order from smallest to largest or vice-versa is?
rank
Data in order from the smallest to largest or vice versa?
range
Data in order from the smallest to largest or vice-versa?
range
Let G be a finite cyclic group prove that every subgroup H of G is isomorphic to some factor group G mod K and every factor group G mod K is isomorphic to some subgroup H?
Let ( G ) be a finite cyclic group generated by an element ( g ). Any subgroup ( H ) of ( G ) can be expressed as ( H = \langle g^k \rangle ) for some divisor ( k ) of the order of ( G ). The factor group ( G/K ) for some subgroup ( K ) is also cyclic, and by choosing ( K ) appropriately (for instance, ( K = \langle g^m \rangle ) where ( m ) divides the order of ( G )), we can ensure that ( G/K ) is isomorphic to ( H ). Thus, every subgroup ( H ) corresponds to a factor group ( G/K ) and vice versa, establishing the desired isomorphism relationships.
Data in order from smallest to largest or vice-versa is called?
Data in order from smallest to largest or vice-versa is called numerical order. It is a systematic arrangement of numbers.
Data in order from smallest to largest or vice-versa?
numerical order
Why are there police men in the world?
Because the bad people of the world will always need someone to prove themselves on -and vice versa!
Data in order from smallest to largest or vice-versa is?
rank
Data in order from the smallest to largest or vice versa?
range
Data in order from the smallest to largest or vice-versa?
range
Data in order from smaller to largest or vice versa?
The data are sorted.
What is Data in order from the smallest to largest or vice a versa?
It is ordered data.
What is the data in order from smallest to largest or vice versa?
It is ordered data.
What is data in order from smallest to largest or vice-versa?
Range what about ranking?
What is data in the order from smallest to largest or vice-versa?
Ordered data.
