Phys 509 - Problem Set 6

Abstract

We show that CG ({g}) is a subgroup CG ({g}) ≤ G. • Closure: For x, y ∈ GG ({g}), then xg = gx and yg = gy, so in particular gxy = xgy = xyg, so xy ∈ CG ({g}) • Inverses: For x ∈ CG ({g}) ⊆ G, ∃x−1 ∈ G s.t. xx−1 = x−1x = e, since G is a group. For g−1 ∈ G exists, then we can write gx−1g−1x = g (gx) x = g (xg) x = gg−1x−1x = e, which shows gx−1 = x−1g, so x−1 ∈ CG ({g}) • Identity: For e ∈ G, eg = ge = g by de nition, so e ∈ CG ({g}). • Associativity: Inherited from G. Part (c) Centralizer

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Cite this paper

@inproceedings{Liu2009Phys5, title={Phys 509 - Problem Set 6}, author={Yun Liu}, year={2009} }