Normal Subgroups Test A normal subgroup of a group G is a subgroup N of G such that every element in G is the product of elements in N. In other words, N is...
Normal Subgroups Test
A normal subgroup of a group G is a subgroup N of G such that every element in G is the product of elements in N. In other words, N is closed under the group operation.
The Normal Subgroups Test determines whether a subgroup is normal or not. It is a simple test to perform, and it provides valuable information about the structure of a group.
How to Perform the Test:
Choose any element a in G.
Show that a is an element of N.
For every element b in N, show that ab is an element of N.
Conclusion: If a is in N, then N is normal.
Examples:
The subgroup of the group Z4 consisting of the elements 0, 1, 2, and 3 is normal, since it is closed under the addition and multiplication of elements.
The subgroup of the group GL(2, R) consisting of all invertible 2x2 matrices with real entries is normal, since it is closed under the matrix multiplication operation.
The group Z6 is not normal, since it is not closed under the addition operation.
The Normal Subgroups test is a powerful tool for determining the structure of groups. It is a simple test to perform, but it provides valuable information about the group, such as its normal subgroups and homomorphisms