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new group theory symmetry group geometric group theory transformation group mathematical action abstract algebra cube symmetry

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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⣴⣴⢶⡆⢠⣶⣷⣦⣦⣆⡶⡇⣻⣷⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⠁⠀⠉⠉⠁⠉⠈⠀⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀ ⢀⠀⣀⣀⣀⣀⣀⣠⣀⢀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠁⠀⠉⠀⠁⠉⠉⠉⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠈⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠙⢶⣄⡀⠀⠀⠀⣠⡴⠛⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠈⠛⣦⣴⠞⠉⠀⠀⠀⠀⠉⠢⣀⠀⠀⠀⠀⠀ ⠀⠀⢀⠴⠊⠉⠻⢦⣀⠀⠀⢀⣤⠞⠉⠉⠳⢦⣀⠀⠀⢀⡴⠞⠉⠑⠢⣀⠀⠀ ⢠⣖⠁⠀⠀⠀⠀⠀⢉⣷⣾⡋⠀⠀⠀⠀⠀⠀⢉⣷⣾⡋⠀⠀⠀⠀⠀⠀⢑⡆ ⠀⠉⠳⢤⡀⠀⣠⡶⠋⠁⠀⠙⠲⣄⡀⢀⣠⠶⠋⠁⠈⠙⠶⣄⡀⢀⣠⠞⠉⠀ ⠀⠀⠀⠀⠙⡿⣍⡀⠀⠀⠀⠀⠀⣨⡿⠿⣅⠀⠀⠀⠀⠀⠀⣨⢿⡏⠁⠀⠀⠀ ⠘⠦⣄⠀⠀⡇⠀⠙⠦⣄⣀⡴⠞⠁⠀⠀⠈⠻⢦⣄⣀⠴⠋⠁⢸⡇⠀⣀⡴⠃ ⠀⠀⠈⠑⢦⣇⠀⠀⠀⠈⡟⢦⣀⠀⠀⠀⠀⢀⡴⢿⡇⠀⠀⠀⢸⡧⠞⠁⠀⠀ ⢠⡀⠀⠀⠀⡏⠳⢤⡀⠀⡇⠀⠉⠳⢦⣠⠞⠉⠀⢸⡇⢀⣠⠖⢹⡇⠀⠀⢀⡀ ⠀⠉⠲⣄⠀⡇⠀⠀⠉⠲⣧⡀⠀⠀⢸⡇⠀⠀⠀⣼⡷⠋⠀⠀⢸⡇⣠⠖⠋⠁ ⠀⠀⠀⠀⠙⡷⣀⠀⠀⠀⡇⠙⠦⣄⢸⡇⣀⡴⠋⢹⡇⠀⠀⣀⢼⡏⠁⠀⠀⠀ ⠈⠢⢀⠀⠀⡇⠈⠑⠦⣀⡇⠀⠀⠈⢹⡟⠁⠀⠀⢸⣇⡴⠚⠁⢸⡇⠀⡀⠔⠀ ⠀⠀⠀⠁⠢⢇⠀⠀⠀⠈⡷⢦⡀⠀⢸⡇⠀⢀⡤⢾⡏⠀⠀⠀⢸⠇⠊⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠐⠄⡀⠀⡇⠀⠉⠲⢼⣧⠶⠋⠀⢸⡇⢀⠠⠂⠁⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢇⡀⠀⠀⢸⡇⠀⠀⠀⡸⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⢸⡇⡀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
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H⊂G
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⚛️➡️{🔵🟢🔴}
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🏛️⟨∗𝑒⟩
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{•••}⚙️
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G≅H
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🔵+🟢→{🟣}
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🚶···🚶⚙️
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new algebraic structures axioms operations definitions foundations abstract algebra modern algebra group theory proofs
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