Center and Centralizers

Abstract Algebra · Axiom Academy

LESSON Center and Centralizers Exploring which elements commute in a group: the center Z(G) and centralizers C(a). Understanding commutativity through visual demonstrations and concrete examples. The center is always a subgroup of G . If Z(G) equals G , the group is abelian (everything commutes). If Z(G) contains only the identity, the group has "no central elements" beyond the trivial one. C(a) is always a subgroup of G Z(G) is the intersection of all centralizers: If a is in the center, then C(a) = G The dihedral group D_4 is the symmetry group of a square, with 8 elements: 4 rotations and 4 reflections. Rotations: e, r, r^2, r^3 (rotations by 0°, 90°, 180°, 270°) Reflections: s, t, u, v (reflections through vertical, horizontal, and two diagonal axes) Computing Z(D₄): We check which elements commute with everything: e (identity) always commutes ✓ r^2 (180° rotation) commutes with all elements ✓ All reflections and 90° rotations do not commute with everything ✗ 4. Example: Quaternion Group Q₈ The quaternion group Q_8 has 8 elements: 1 (identity) commutes with everything ✓ -1 commutes with everything (since (-1)x = x(-1) ) ✓ i, j, k do not all commute with each other ✗ Which elements commute with i ? Consider , the group of invertible 2×2 matrices over . A matrix A is in the center if it commutes with all matrices. This happens if and only if A is a scalar matrix: These are the multiples of the identity matrix! Scalar matrices commute with everything

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