If G is a group of order |G| = 35
ID: 2985210 • Letter: I
Question
(a) Explain why G is a direct product Z7 × Z5, or is one of the possible semidirectproducts Z7×Z5.
(b) Determine all possible homomorphisms Φ : Z5 → Aut(Z7, +) ∼= (U7, ·).
(c) Make a complete list of the distinct isomorphism types of all groups of order |G| = 35. (No two groups in your list should be isomorphic.) Where does the well known group Z35 of order 35 appear in your list? (a) Explain why G is a direct product Z7 × Z5, or is one of the possible semidirect
products Z7×Z5.
(b) Determine all possible homomorphisms Φ : Z5 → Aut(Z7, +) ∼= (U7, ·).
(c) Make a complete list of the distinct isomorphism types of all groups of order |G| = 35. (No two groups in your list should be isomorphic.) Where does the well known group Z35 of order 35 appear in your list?
Explanation / Answer
Notice that 35 = 5*7 and by Sylow Theorems the number of Sylow 5 subgroups is 1 (call it K) and the number of Sylow 7 subgroups is 1 (call it H). Hence, they are normal. Now the intersection of H and K has to be 1 because no element has order 5 and order 7 at the same time.
So G=HK = H x K. Since the only group of order 5 is Z5 and the only one of order 7 is Z7, G = Z5 x Z7. So there is actually only one group of order 35 and it is this one.