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.
- 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
- 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
- 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
- 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
