By O.U. Schmidt

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After the outbreak of the Franco– Prussian War in July 1870, he left for Italy but was arrested in Fontainebleau on suspicion of being a German spy and was only released due to the intervention of the French mathematician Jean Gaston Darboux (1842–1917). Returning to Norway, he was awarded a doctorate by the University of Christiania in July 1872, for a thesis entitled On a class of geometric transformations, and subsequently appointed to a chair. Over the next few decades he made many important contributions to geometry and algebra, many in collaboration with Klein and Friedrich Engel (1861– 1941).

Then H = ( H, ∗ H ) is a subgroup of G (written H < G) if and only if: SG1 H is closed under the action of ∗ H . That is, for all h1 , h2 ∈ H, the product h1 ∗ H h2 ∈ H too. SG2 For all h ∈ H, the inverse h−1 ∈ H as well. 2 Let G = ( G, ∗) be abelian, and let H < G be a subgroup of G. Then H is also abelian. The converse doesn’t hold, however, since nonabelian groups can have abelian subgroups. For example, we noted earlier that D3 has a subgroup isomorphic to Z3 which is therefore abelian, but D3 itself isn’t abelian.

A bit of examination or experimentation shows that these matrices represent reflections in lines passing through the origin: C is reflection in the x–axis, D is √ √ y=0 C reflection in the line y = − 3x, and E is reflection in the line y = 3x. This matrix group tells us how six specific geometric transformations interact with each other. 5: Reflections represented by same as an anticlockwise rotation through an angle of 3 . the matrices C, D and E m2 m3 m1 This geometric approach is a useful and illuminating way of thinking about groups.