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Mathematics — Unit 1 — Test 18

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Cutoff (GEN): 70%

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1

In a finite p‑group (|G|=p^n), the intersection of all maximal subgroups is

A

the Frattini subgroup Φ(G)

B

trivial

C

equal to G

D

the center Z(G)

2

For a finite group G, if every proper subgroup is cyclic, then G is

A

dihedral

B

cyclic

C

either cyclic or isomorphic to Q8

D

simple

3

Let φ:G→H be a homomorphism. Which is always true?

A

im φ ≅ H/ker φ

B

G ≅ H

C

ker φ ≅ H/im φ

D

im φ ≅ G/ker φ

4

In S6, how many 3‑cycles are there?

A

40

B

60

C

20

D

80

5

Let G be cyclic of order 100. How many elements have order dividing 20?

A

20

B

40

C

12

D

30

6

Let G act transitively on a set Ω with |Ω|=n. The index |G:G_α| equals

A

n

B

|G|

C

divides n

D

n!

7

Which implication always holds in an integral domain?

A

both directions

B

irreducible ⇒ prime

C

prime ⇒ irreducible

D

neither direction

8

In a PID, every nonzero prime ideal is

A

not necessarily principal

B

maximal

C

nilpotent

D

principal but not maximal

9

Z[x] is

A

not a UFD

B

a field

C

a UFD but not a PID

D

a PID

10

Over a field F, F[x] is

A

a UFD but not PID

B

a Euclidean domain

C

both PID and Euclidean domain

D

a PID

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#1	User 0	20%	September 14, 2025

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