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I have generators for my group in GAP. Additionally, I possess a specific unitary element and wish to determine whether this unitary element arises from the Lie algebra of my group's generators. Initially, I considered a brute-force approach, intending to compute the matrix exponential of a particular generator of the group. However, I couldn't find any matrix exponential function in GAP (e.g., expm, ExpMat, MatExp, etc.). Now, I have two questions:

  1. How can I compute the matrix exponential in GAP?
  2. Is there a systematic method to compute the Lie algebra of the group generators, allowing me to identify my specific unitary element, and subsequently terminate the computation in GAP?
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