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Abstract Algebra

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Moderate

Testing for Normal Subgroups

ABSALG-YEFTY@

In which of the following pairs is $N$ NOT a normal subgroups of $G$? Select ALL that apply.

A

$N$ is the subgroup of $2\times 2$ matrices with real coefficients of determinant $1$ and $G$ is the group of all invertible $2\times 2$ matrices with real coefficients under matrix multiplication.

B

$H=\langle (123)\rangle$ is the subgroup of $G=A_4$, the alternating group on four symbols.

C

$H$ is the subgroup of nonconstant linear functions and $G$ is the group of all invertible functions from $\mathbb{R}$ to $\mathbb{R}$ under the composition of functions.

D

$H$ is the subgroup of elements $x$ in the Abelian group $G$ such that $x^2=e$.