0
To prove that the residue classes modulo a prime ( p ) form a multiplicative group, consider the set of non-zero integers modulo ( p ), denoted as ( \mathbb{Z}_p^* = { 1, 2, \ldots, p-1 } ). This set is closed under multiplication since the product of any two non-zero residues modulo ( p ) is also a non-zero residue modulo ( p ). The identity element is ( 1 ), and every element ( a ) in ( \mathbb{Z}_p^* ) has a multiplicative inverse ( b ) such that ( a \cdot b \equiv 1 \mod p ) (which exists due to ( p ) being prime). Thus, ( \mathbb{Z}_p^* ) satisfies the group properties of closure, associativity, identity, and inverses, confirming it is a multiplicative group.
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What are the elements in 2x2 matrix with integer modulo 3 which is a group?
48
What is the order of an element in a group?
The order of an element in a multiplicative group is the power to which it must be raised to get the identity element.
What is the additive inverse of 9.1?
We are talking group theory here. A group with addition has an additive inverse. A group with multiplication has a multiplicative inverse. The additive inverse of a number x is a y with x + y = 0. The additive inverse of x is written -x. Hence, the additive inverse of 9.1 equals -9.1. The reason that this question can arise is that beyond groups, there are rings and fields. Rings and fields have, besides addition, also multiplication. An element can have an additive inverse and a multiplicative inverse at the same time.
What do you understand by expansion through integration Differentiate between vertical and horizontal integration giving examples?
integration or social integration is the uniting of formerly separate group with the obliteration of any previous social and culture group diference. its is a constant factor in a social development. it brodends the social group by uniting with other groups. vertical integrance is the uniting of lower class group with the upper or the uniting of upper class group with lower class group... for examples as we see in the mosque different people from different classes unite together without discrimating the class.... horizental intergration is that in which ones social position changes integration or social integration is the uniting of formerly separate group with the obliteration of any previous social and culture group diference. its is a constant factor in a social development. it brodends the social group by uniting with other groups. vertical integrance is the uniting of lower class group with the upper or the uniting of upper class group with lower class group... for examples as we see in the mosque different people from different classes unite together without discrimating the class.... horizental intergration is that in which ones social position changes
How do you prove that the group has no subgroup of order 6?
To prove that a group ( G ) has no subgroup of order 6, we can use the Sylow theorems. First, we note that if ( |G| ) is not divisible by 6, then ( G ) cannot have a subgroup of that order. If ( |G| ) is divisible by 6, we analyze the number of Sylow subgroups: the number of Sylow 2-subgroups ( n_2 ) must divide ( |G|/2 ) and be congruent to 1 modulo 2, while the number of Sylow 3-subgroups ( n_3 ) must divide ( |G|/3 ) and be congruent to 1 modulo 3. If both conditions cannot be satisfied simultaneously, it implies that no subgroup of order 6 exists.
How many composition series of group z modulo 30?
4
What are the elements in 2x2 matrix with integer modulo 3 which is a group?
48
What group of real numbers will 9.34 go to?
A group containing 9.34 is a set of numbers, with some operation defined on the set that also satisfies:closure,associativity,identity, andinvertibility.Two simple groups will be the additive group of 9.34 and all its multiples (including negative ones). The identity is 0.The other is the multiplicative group consisting of all powers of 9.34 and the identity is 1.There can be a finite additive group derived from the first by defining the operation as modulo addition, and similarly with the multiplicative group.Finally, any group that contains one of these groups and also maintains the four conditions listed above, for example, all rational numbers, will also meet the requirements.
What is the smallest number of a group?
You can have a Group with only one element: 0 if it an additive group or 1 if it is multiplicative.
What is the order of an element in a group?
The order of an element in a multiplicative group is the power to which it must be raised to get the identity element.
When is thirty plus thirty equal to one?
minutes and hoursWhen the group you consider is (Z59, +) or you are working modulo 59.
What generalization for each group of numbers that is not true for the other group of numbers 1 7 4 and 6 3 9?
All the numbers in each group have the same modulo 3 value.
Is group and class the same thing?
Yes a group and a class are the same thing.
What is a set of numbers at fixed distances?
In the branch of algebra called group theory, they are called equivalent classes or residual classes. They are generated with the fixed distance as the basis of the modulo relationship.
What is a group of students working together?
What a Group of Students is CalledA group of students is called a class and is often referred to as (name of teacher)'s class to distinguish one class from another. However, the generalized term 'gaggle' is also applied to a group of students in the comic strip Piled Higher and Deeper by Jeorge Cham, PhD.
What is a group of children called?
A severe pain in the neck
What is the analogy for soldier is to squad as pupil is to?
class
