117 Simple Group Names: Unraveling the Complexity of Abstract Algebra

117 Simple Group Names: Unraveling the Complexity of Abstract Algebra

The world of abstract algebra is vast and intricate, filled with complex concepts and intricate theories. Within this realm, the study of simple groups holds a prominent position, captivating the minds of mathematicians for centuries. These groups, characterized by their lack of normal subgroups other than the trivial ones, offer a glimpse into the fundamental structure of algebraic systems. Delving into the world of simple groups reveals a fascinating array of names, each carrying a story of discovery, mathematical insight, and the collective efforts of brilliant minds.

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  1. Finite Simple Groups:

    • A5: The alternating group on 5 elements
    • PSL(2, 5): The projective special linear group of degree 2 over the field with 5 elements
    • M11: The Mathieu group on 11 elements
    • M22: The Mathieu group on 22 elements
    • J1: The Janko group on 100 elements
    • J2: The Janko group on 175 elements
    • J3: The Janko group on 248 elements
    • Co1: The Conway group on 23 elements
    • Co2: The Conway group on 22 elements
    • Co3: The Conway group on 21 elements

  2. Sporadic Simple Groups:

    • M23: The Mathieu group on 23 elements
    • M24: The Mathieu group on 24 elements
    • HS: The Higman-Sims group
    • McL: The McLaughlin group
    • He: The Held group
    • Ru: The Rudvalis group
    • Suz: The Suzuki group
    • O’N: The O’Nan group
    • F5: The Fischer group on 248 elements
    • F3: The Fischer group on 220 elements
    • F2: The Fischer group on 168 elements

  3. Lie Type Simple Groups:

    • A1: The special linear group of degree 2 over the field with 2 elements
    • B2: The special orthogonal group of degree 3 over the field with 2 elements
    • C3: The symplectic group of degree 2 over the field with 3 elements
    • D4: The orthogonal group of degree 4 over the field with 2 elements
    • E6: The exceptional Lie group of type E6
    • E7: The exceptional Lie group of type E7
    • E8: The exceptional Lie group of type E8
    • F4: The exceptional Lie group of type F4
    • G2: The exceptional Lie group of type G2

  4. Other Simple Groups:

    • PSL(2, q): The projective special linear group of degree 2 over the field with q elements, where q is a prime power
    • PGL(2, q): The projective general linear group of degree 2 over the field with q elements, where q is a prime power
    • PSU(3, q): The projective special unitary group of degree 3 over the field with q elements, where q is a prime power
    • PSU(4, q): The projective special unitary group of degree 4 over the field with q elements, where q is a prime power
    • Sz(q): The Suzuki group over the field with q elements, where q is a prime power
    • U(q): The unitary group of degree q over the field with q elements, where q is a prime power
    • Sp(2n, q): The symplectic group of degree 2n over the field with q elements, where q is a prime power
    • O(n, q): The orthogonal group of degree n over the field with q elements, where q is a prime power

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