In each of the following cases, determine whether or not $G$ is isomorphic to the product group $H \times K$.

Question

In each of the following cases, determine whether or not $G$ is isomorphic to the product group $H \times K$.

a) $G $= {invertible upper triangular $2\times2$ matrix },$H=${invertible diagonal matrices} $K$={upper triangular matrices with diagonal entries $1$}

b)$G = \mathbb{C}^{\times}$, $H$ = {unit circle} , $K$= {postive real number}

My attempt : in both a) and b) cases $G$ is not isomorphic to the product group $H \times K$.

i was thinking about first theorem of isomorphism

For $a)$ $\frac{G}{H}$ not isomorphics to $K$, because H is not normal subgroup of $G$, so $ G \neq H \times K$

for $b)$ $\frac{G}{K} =\frac{\mathbb{C}^{\times}}{\text{postive real number}} $will isomorphic to $\{ -1,+1\}$, that is it $\{-1,+1\}$ $\neq$ unit circle, so $ G \neq H \times K$

Is it true ??

any hints/solution will be appreciated

thanks u

Answer 1

$a)$ is False ,take

$ \begin{bmatrix} 2 &0 \\ 0& 1 \end{bmatrix}\begin{bmatrix} 1 &2 \\ 0& 1 \end{bmatrix} \neq \begin{bmatrix} 1 &2 \\ 0& 1 \end{bmatrix}\begin{bmatrix} 2 &0 \\ 0& 1 \end{bmatrix} $.

$b)$ is True , since $\mathbb{C^×}$ is abelian, both K and H are normal subgroups. Moreover the two subgroups intersect only in the identity of $\mathbb{C}^{\times}$.The product $KH$ is the whole $\mathbb{C}$ follows from the fact that every complex number has an expression in polar coordinates, that is as a product of a positive real number and a number in the unit circle