An element is called a square if it can be expressed in the...

An element is called a square if it can be expressed in the form $b^{2}$ for some $b$. Suppose that $G$ is an Abelian group and $H$ is a subgroup of $G$. If every element of $H$ is a square and every element of $G / H$ is a square, prove that every element of $G$ is a square. Does your proof remain valid when "square" is replaced by " $n$ th power" where $n$ is any integer?

Key Concepts

Abelian Group

An Abelian group is a group in which the group operation is commutative. This commutativity property simplifies many arguments in group theory, particularly when dealing with products and powers of elements, since the order of multiplication does not affect the result.

Subgroup and Quotient Group

A subgroup is a subset of a group that itself forms a group under the same operation, while a quotient group is formed by partitioning the group into cosets of a normal subgroup. These concepts are fundamental in analyzing the structure of groups and in transferring properties from subgroups or quotient groups back to the entire group.

Power Maps in Groups

The power map in a group refers to the function that sends each element to its nth power, where n is an integer. Studying whether every element of a group can be expressed as an nth power (or square in the special case) is important in understanding the structure and behavior of the group, as well as in exploring properties like surjectivity of these maps.

Lifting Properties from Quotient Groups to Groups

A key technique in group theory involves proving that a property holding in both a subgroup and the corresponding quotient group can be 'lifted' to the whole group. This method often uses the fact that every element in the quotient can be represented in a certain way, combined with properties satisfied by the subgroup, to show that the entire group also satisfies the desired property.

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