The $T_z$~=~$-frac{3}{2}$ nucleus ${}^{21}$Mg has been studied by Coulomb excitation on ${}^{196}$Pt and ${}^{110}$Pd targets. A 205.6(1)-keV $gamma$-ray transition resulting from the Coulomb excitation of the $frac{5}{2}^+$ ground state to the first excited $frac{1}{2}^+$ state in ${}^{21}$Mg was observed for the first time. Coulomb excitation cross-section measurements with both targets and a measurement of the half-life of the $frac{1}{2}^+$ state yield an adopted value of $B(E2;frac{5}{2}^+rightarrowfrac{1}{2}^+)$~=~13.3(4)~W.u. A new excited state at 1672(1)~keV with tentative $frac{9}{2}^+$ assignment was also identified in ${}^{21}$Mg. This work demonstrates large difference of the $B(E2;frac{5}{2}^+rightarrowfrac{1}{2}^+)$ values between $T$~=~$frac{3}{2}$, $A$~=~21 mirror nuclei. The difference is investigated in the shell-model framework employing both isospin conserving and breaking USD interactions and using modern textsl{ab initio} nuclear structure calculations, which have recently become applicable in the $sd$ shell.
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