A course in abstract algebra by Nicholas Jackson

By Nicholas Jackson

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The symmetric group S4 is isomorphic to the symmetry group of the tetrahedron. Later on we will meet Cayley’s Theorem, which states that any group can be regarded as a group of permutations (although not, in general, the full symmetric group Sn for some value of n). To investigate these permutation groups, we need a coherent and consistent notation, at least for permutations on finite sets. One method, given that a permutation σ : X → X is determined completely by its action on the elements of the set X, is to represent it in the form of an array: x1 σ ( x1 ) x2 σ ( x2 ) ...

9 Let H, K < G be subgroups of some group G. Then their intersection H ∩ K is also a subgroup of G, and of H and K. Proof Since both H and K are subgroups of G, they both contain the identity e of G, and hence their intersection H ∩ K does as well, and is therefore not empty. Given elements g1 , g2 ∈ H ∩ K, it follows that both g1 , g2 ∈ H and g1 , g2 ∈ K. Therefore their product g1 ∗ g2 ∈ H and also g1 ∗ g2 ∈ K, hence g1 ∗ g2 ∈ H ∩ K. Thus H ∩ K is closed under the induced multiplication operation.

The full isometry group E3 = Isom(R3 ) also includes reflections in some plane, glide reflections (reflections in a plane followed by a translation parallel to that plane), roto-reflections (rotation about an axis followed by reflection in a plane perpendicular to that axis) and inversion in a point (in which every point in R3 is mapped to an equidistant point on the other side of the chosen fixed point). This group E3 is the three-dimensional Euclidean group. Wikimedia Commons / User:Cyp Just as we obtained interesting examples of groups by restricting ourselves to elements of E2 or E2+ which mapped some subset of R2 to itself, it turns out that we can obtain further interesting examples by doing something similar with the groups E3 and E3+ .

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