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murmillo
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murmillo said:OK, this following paragraph is where I'm stuck: Since K is normal, yxy^-1 is an element of K, a power of x, say x^i, where i is between 1 and 7. So the elements x and y satisfy the relations x^7=1, y^3=1, yx=(x^i)y. OK, this makes sense. But, says Artin, the relation y^3 restricts the possible exponents i:
x=(y^3)(x)(y^-3) = (y^2)(x^i)(y^-2) = y(x^(i^2))(y^-1)=x^(i^3)).
An isomorphism class is a set of groups that are considered equivalent under the concept of isomorphism. In other words, they have the same structure and can be mapped onto each other in a way that preserves the group operations.
There is only one isomorphism class for groups of order 21. This is because any two groups of order 21 are isomorphic to each other, meaning they have the same structure and can be mapped onto each other in a way that preserves the group operations.
The group in the isomorphism class of groups of order 21 is called the cyclic group of order 21, denoted by C21. This group is generated by a single element and has 21 elements in total.
The cyclic group of order 21, C21, has the structure of a group with a single generator, denoted by x, and the following operation rule: x^0 = e (identity element), x^1 = x, x^2 = x^3 = ... = x^20 = e. This means that any element in the group can be expressed as a power of x.
The cyclic group of order 21, C21, has the following properties: it is abelian (commutative), it is a finite group, it has exactly one subgroup of every divisor of 21, and it is a cyclic group, meaning it is generated by a single element. Additionally, it is also a simple group, meaning it has no nontrivial normal subgroups.