## Subgroups

Let $$G$$ be a group and $$H$$ any subset of $$G$$. Let $$a,b$$ be any two elements of $$H$$. Now… Click here to read more

From basic to higher mathematics

Let $$G$$ be a group and $$H$$ any subset of $$G$$. Let $$a,b$$ be any two elements of $$H$$. Now… Click here to read more

The necessary and sufficient conditions for a subset of a group to be a subgroup are stated in the following… Click here to read more

Theorem 1: The intersection of two subgroups of a group $$G$$ is a subgroup of $$G$$. Proof: Let $${H_1}$$ and… Click here to read more

Theorem 1: Every subgroup of a cyclic group is cyclic. Proof: Let $$G = \left\{ a \right\}$$ be a cyclic… Click here to read more

If $$G$$ is a group, $$H$$ is a subgroup and $$a$$ is any element in $$G$$, then the set \[\left\{… Click here to read more

Let $$H$$ be a subgroup of group $$G$$. We know that no right coset of $$H$$ in $$G$$ is empty… Click here to read more

Theorem 1: If $$h \in H$$, then the right (or left) coset $$Hh$$ or $$hH$$ of $$H$$ is identical to… Click here to read more

Let $$H$$ be a subgroup of a group $$G$$. If the element $$a$$ of $$G$$ belongs to the right coset… Click here to read more

Lagrange’s Theorem The order of a subgroup of a finite group divisor of the order of the group. Proof: Let… Click here to read more

Let us consider the set of all complexes of a group $$G$$, which is nothing but a power set of… Click here to read more

Example 1: Find the proper subgroups of the multiplicative group $$G$$ of the sixth roots of unity. Solution: From trigonometry… Click here to read more