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Is a group of order 24 abelian group or not?
The abelian groups of order 24 are C3xC8, C2xC12, C2xC2xC6. There are other 12 non-abelian groups of order 24
How to determine number of isomorphic group of order 121?
Since 121 is the square of a prime, there are only two distinct isomorphic groups.
How many abelian groups up to isomorphism are there of order 32?
7 groups, use the structure theorem
Can a non-abelian group have a torsion subgroup?
Yes, a non-abelian group can have a torsion subgroup. A torsion subgroup is defined as the set of elements in a group that have finite order. Many non-abelian groups, such as the symmetric group ( S_3 ), contain elements of finite order, thus forming a torsion subgroup. Therefore, the existence of a torsion subgroup is not restricted to abelian groups.
If PS are distinct primes show that an abelian group of order PS must be cyclic?
If ( G ) is an abelian group of order ( PS ), where ( P ) and ( S ) are distinct primes, then by the Fundamental Theorem of Finite Abelian Groups, ( G ) can be expressed as a direct product of cyclic groups of prime power order. The possible structures for ( G ) are ( \mathbb{Z}/PS\mathbb{Z} ) or ( \mathbb{Z}/P^k\mathbb{Z} \times \mathbb{Z}/S^m\mathbb{Z} ) with ( k ) and ( m ) both ( 1 ). However, since ( P ) and ( S ) are distinct primes, the only way for ( G ) to maintain order ( PS ) while being abelian is for it to be isomorphic to ( \mathbb{Z}/PS\mathbb{Z} ), which is cyclic. Thus, ( G ) must be cyclic.
What is the number of groups of order 8 upto isomorphisms?
There are 5 groups of order 8 up to isomorphism. 3 abelian ones (C8, C4xC2, C2xC2xC2) and 2 non-abelian ones (dihedral group D8 and quaternion group Q)
How do you determine number of isomorphic groups of order 10?
There are two: the cyclic group (C10) and the dihedral group (D10).
Prove that a group of order three is abelian?
By LaGrange's Thm., the order of an element of a group must divide the order of the group. Since 3 is prime, up to isomorphism, the only group of order three is {1,x,x^2} where x^3=1. Note that this is a finite cyclic group. Since all cyclic groups are abelian, because they can be modeled by addition mod an integer, the group of order 3 is abelian.
Why D2 and z12 are not isomorphic?
D2, the dihedral group of order 4, consists of rotations and reflections of a square, while Z12, the cyclic group of order 12, is generated by the addition of integers modulo 12. D2 is not cyclic, as it cannot be generated by a single element, whereas Z12 is cyclic, generated by 1. Furthermore, the structure of the groups differs: D2 has elements of order 2 (the reflections) and elements of order 4 (the rotations), while Z12 has elements of various orders that are consistent with a cyclic structure. Hence, their different algebraic structures confirm that D2 and Z12 are not isomorphic.
Why is non abelion group smallest of six elements in mathematics?
The non-abelian group of smallest order (six elements) is the symmetric group S3. This group consists of all possible permutations of three elements, and it is non-abelian because the composition of permutations does not commute in general. It is the smallest non-abelian group because any group with fewer than six elements is either abelian or not a group.
Is the decimal number 2.5 odd or even?
It can't be either, because the rationals aren't order isomorphic to the integers.
What property states that numbers can be added or multiplied in any order without affecting the answer?
The commutative or Abelian property.
