Paper 3, Section I, E

Groups | Part IA, 2017

What does it mean to say that HH is a normal subgroup of the group GG ? For a normal subgroup HH of GG define the quotient group G/HG / H. [You do not need to verify that G/HG / H is a group.]

State the Isomorphism Theorem.


G={(ab0d)a,b,dR,ad0}G=\left\{\left(\begin{array}{ll} a & b \\ 0 & d \end{array}\right) \mid a, b, d \in \mathbb{R}, a d \neq 0\right\}

be the group of 2×22 \times 2 invertible upper-triangular real matrices. By considering a suitable homomorphism, show that the subset

H={(1b01)bR}H=\left\{\left(\begin{array}{ll} 1 & b \\ 0 & 1 \end{array}\right) \mid b \in \mathbb{R}\right\}

of GG is a normal subgroup of GG and identify the quotient G/HG / H.

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