I am new in linguistics and I am an ESL student. When I check dictionaries such as Merriam-Webster, Random House Webster, Webster’s New world college, American Heritage, Cambridge dictionary etc and listen to the pronunciation, there are different division of syllable in a word. For example: Other /ʌ ðər/ /ʌð ər/ /ʌð ðər/ , cabin /kæ bɪn/ /kæb ɪn/ /kæb bɪn/. I have known the Maximal Onset Principle and Sonority Scale, but I am not sure about that. Is there a certain rule for dividing syllable in a word.
Like most everything in linguistics, the answer depends on the theory being assumed. It is generally but not universally assumed that segments are assigned to syllables in a definite way, in every language. The three syllabifications that you encountered for "other, cabin" cover the three main theories of English for two-syllable sequences with a short stressed vowel followed by an unstressed vowel, and a single consonant in the middle. The one proviso is that [ʌð ðɹ̩] is a transcriptional artifact because you cannot place a syllable boundary in the middle of the consonant – the theory is that the consonant is both in the coda of the preceding syllable and the onset of the following syllable (not that there are two consonant segments).
Some theories do not allow a segment to be in two different syllables, so such a theory would not accept [kæb bɪn]. The syllabification /kæ.bɪn/ has the problem that in English, syllables don't end with short stressed vowels: except, in words like "cabin". Another way to state the generalization about short stressed vowels is that words don't end in them (*[hɪ, ðɛ, mæ]). One often hears the objection that "feh, meh" are not un-utterable by all English speakers. On the other hand, hyper-slow speech syllabification judgments from speakers tends towards [kæ.....bɪn].
The syllabification [kæb.ɪn] satisfies the theoretical desideratum that stressed syllables should be heavy, but violates the theoretical desideratum that syllables should have onsets if possible (this is a point in favor of [kæb.bɪn] – the "ambisyllabicity" analysis). It also has fewer problems with "hangar" since the claim is that this is [hæŋ.ɹ̩] with /ŋ/ in the coda (words cannot start with /ŋ/, which is generalized to the claim that syllables cannot begin with /ŋ/).
The rules for aspiration, flapping and glottalization are involved in the syllabification debate. The distribution of [t, tʰ, ɾ, t̚] is in part related to syllable position, and each theory of syllabification has its particular account of where consonants are aspirated. The VC.CV theory relies on restricting the rule of aspiration to consonants that are "not syllable-final". The V.CV theory crucially requires that the syllable getting aspiration must be stressed (so in "later", there is no aspiration because the second syllable is unstressed). A counter-argument is that aspiration also applies in unstressed syllables ("tomato, potato, corona"), and the counter-counter-claim is "No it's not, these are unaspirated). VOT i.e. aspiration is a continuum and in "tomato" VOT is shorter than it is in "tuba", so you could say that /t/ in "tomato" is unaspirated, but in "stop" isn't even more unaspirated. (I don't buy that claim, so I personally reject the pure V.CV claim).
This matter has been debated for decades, and there has not been a definitive resolution to the question. A further entry into the contest is the notion of the "foot", a grouping of syllables with the stressed syllable on the left and following unstressed syllables thereafter. Under than analysis, the restriction on /ŋ/ or where you get aspiration (also /h/ as a segment) is stated in terms of feet – aspiration and /h/ cannot be in the middle of the foot, /ŋ/ cannot be at the beginning of a foot.