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ZixuanZhang
ZixuanZhang
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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 .

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