Isospin symmetry in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>B</mml:mi><mml:mo>(</mml:mo><mml:mi>E</mml:mi><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math> values: Coulomb excitation study of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mmultiscripts><mml:mi>Mg</mml:mi><mml:mprescripts/><mml:none/><mml:mn>21</mml:mn></mml:mmultiscripts></mml:math>

The ${T}_{z}=\ensuremath{-}\frac{3}{2}$ nucleus $^{21}\mathrm{Mg}$ has been studied by Coulomb excitation on $^{196}\mathrm{Pt}$ and $^{110}\mathrm{Pd}$ targets. A 205.6(1)-keV $\ensuremath{\gamma}$-ray transition resulting from the Coulomb excitation of the ${\frac{5}{2}}^{+}$ ground state to the first excited ${\frac{1}{2}}^{+}$ state in $^{21}\mathrm{Mg}$ 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 $B(E2;{\frac{5}{2}}^{+}\ensuremath{\rightarrow}{\frac{1}{2}}^{+})=13.3(4)$ W.u. A new excited state at 1672(1) keV with tentative ${\frac{9}{2}}^{+}$ assignment was also identified in $^{21}\mathrm{Mg}$. This work demonstrates a large difference in the $B(E2;{\frac{5}{2}}^{+}\ensuremath{\rightarrow}{\frac{1}{2}}^{+})$ value 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 ab initio nuclear structure calculations, which have recently become applicable in the $sd$ shell.