Isomorphic or not?

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Originally Posted by tttcomrader

2. Show that no two of $D_6 \ , \ T \ , \ A_4$ are isomorphic, $T = <a,b : a^6 =1, b^2 = a^3, bab^{-1} = a^{-1} >$

$A_4$ has no element of order six while $D_6$ has and so does $T$ .
While $T$ has a unique subgroup of order six and $D_6$ has three of order six.

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3. For this one I'm just simply lost, would anyone please explain more what H is?

Define $\mathbb{H} = \mathbb{R}^4$ any element of $H$ is written as $(a,b,c,d)$ but we write it as $a+bi+cj+dk$ . Now define $i^2=j^2=k^2=-1$ and $ij=k, jk=i,ki=j,ji=-k,ik=-j,ik=-j$ . With this you extend this operation as if it was distributive. For example, $(1+i+j)(k) = k+ik+jk = k - j + i$ . Note that $(\mathbb{H},\cdot )$ is not commutative.