Is this German sentence correct?

Yes, the sentence in the book is correct.

1) The subject of the sentence is "vier kleine Katzen" (four little cats). You can also use a different word order - the sentence still means the same thing: "Und vier kleine Katzen liegen neben ihr" (And four little cats are lying next to her). Because the subject, "four little cats", is plural, you have to use the plural form of the verb "liegen" (to lie):

Vier kleine Katzen (=sie) liegen neben ihr

Four little cats (=they) lie/are lying next to her

2) "Und neben ihr lag vier kleine Katzen" is wrong. "lag" is the singular form of the verb, e.g. "er lag" = "he lay/was lying". Since the subject (four little cats) is plural, you have to use the plural form of the verb. That would be "sie lagen" (= they lay/were lying). You could say "Und neben ihr lagen vier kleine Katzen" (And four little cats were lying next to her), but that would be past tense. Your original sentence is written in the present tense.

3) "Und neben ihr sind vier kleine Katzen" (And next to her are four little cats) is possible, though it doesn't sound as good as the original sentence.

correct:

"Und neben ihr liegen vier kleine Katzen."

"Und neben ihr liegen vier kleine Katzen."

Sentence is correct.

Ich liege

du liegst

er/sie /es liegt

wir liegen

ihr liegt

SIE LIEGEN

It rather relies upon on what you attempt to declare. Are you attempting to declare the lamp is amazingly previous yet yet it rather is amazingly fantastically and blue. Meine Lampe ist alt, aber sie ist sehr schön und blau. Or are you in simple terms pointing out it rather is previous, effective, and blue becuase if it rather is the case you are able to desire to get holiday of the auch. Meine Lampe ist alt. Sie ist sehr schön und blau.

"liegen" means "lie"