Groups of Order 8
Every group of order is isomorphic to one of , , , , or .
Lemma 6.8 (Groups of Order 8)
If , then is isomorphic to one of , , , , or .
Proof
Lagrange’s theorem restricts element orders to , , , and . An element of order gives , while if every non-trivial element has order , repeated use of the direct product theorem gives . If an element has order , the possible relations with an element outside its generated subgroup split into an abelian case giving , a dihedral case giving , and a case with giving .
Related
Stated in
- Lemma 6.8 (Groups of Order 8)§6 Finite Groups
