All right, this is probably worth a little more explanation. Transitivity is a measure of whether a verb requires a direct object in order to form a grammatically complete sentence. Verbs that do not require or permit a direct object are intransitive. Verbs that require a direct object are transitive (and sometimes ditransitive if they also require a secondary object, such as in indirect object).
However, English (of course) gets a little complicated, because there are some verbs that have more than one meaning / sense (much as "lag" has two different senses -- to fall behind, and to cover pipes with insulation). So there are verbs which have one sense that is intransitive and another sense that is transitive (usually a closely related meaning). For example, "I danced" (intransitive verb) and "I danced my shoes to pieces" (a ditransitive verb with a causative sense that takes a direct object, "my shoes", and a prepositional object, "to pieces"). Such verbs (that have two or more senses with differing transitivity) are sometimes called 'ambitransitive' (like ambidextrous, but more annoying). To give a slightly more common example, "he ate" (describing an activity without focusing on any particular thing that the activity was in relation to) versus "he ate the slice of pie" (describing what he did to a particular thing).
Another complexity is that intransitive verbs can sometimes permit non-direct objects. Which is to say, they won't allow a direct object, but they will allow an indirect object or a prepositional object; yet they don't usually need such a non-direct object to be complete. For example: "I listened" (intransitive verb that forms a complete sentence alone) versus "I listened to the radio" (intransitive verb plus optional prepositional object, "to the radio", that gives focus and direction to the activity of the verb). Or, to give another example, "She laughed" versus "she laughed at the cat".
Now, there are a couple of options with respect to the verb 'to lag'. It might be a pure intransitive verb, a pure transitive verb, an ambitransitive verb, or an intransitive verb that permits non-direct objects. In the sense of "to apply insulation" the verb 'to lag' appears to be transitive, in that you can say "We lagged the pipes" (where "the pipes" is a direct object).
However, in my initial estimation the verb 'to lag' (in the sense of being slower / behind) did not appear to be either transitive or ambitransitive; because it seemed as weird to me to say "I lagged my brother" as "I slept my chair" or "I fell the hole" (and because all the dictionaries I consulted only used lag in conjunction with "behind"). Subsequent research via google appears to show that 'to lag' is becoming popular as a transitive verb in talking about market trends (e.g. "the stock lagged the rest of the technology sector"). It also appears to be becoming popular as a transitive causative verb talking about programs that slow down computers and/or internet connections (e.g. "Windows Vista lagged my computer all to Hell"). However, while these meanings are clearly very closely related, I am not 100% convinced they are the same sense as the verb in "we lagged behind the main group". "We lagged the main group" still sounds wrong to my ears (though perhaps this is just me). So I will continue with the assumption that 'to lag' in that particular sense is intransitive.
Now the question is: is it a pure intransitive verb, or does it permit non-direct objects (like the verb 'to laugh', which allows you to say "I laughed at the cat")? That is to say, in the sentence "we lagged behind the main group", is the phrase "behind the main group" a prepositional object, or an adverbial prepositional phrase (i.e. not a prepositional object)?
Generally speaking, prepositional phrases (PPs) that tell us where the action of the verb occurred are adverbial PPs rather than prepositional objects. For instance, 'to fall' is an intransitive verb, and in the sentence "I fell down the hole" the prepositional phrase tells us where the falling took place (i.e. down the hole). Similarly, "I danced in the park" and "I read at the library" are intransitive (or ambitransitive) verbs with adverbial PPs rather than prepositional objects, because the PPs tell us where the action took place rather than what the action was directed at. Conversely, "I looked at the map" and "I listened to the radio" involve intransitive verbs with prepositional objects, because the PPs tell us what the action of the verb was directed at rather than merely where it occurred.
"I lagged behind the main group" appears to be telling us where the action of the verb occurred (i.e. behind the main group), instead of telling us what the action was directed at. Hence, I think 'to lag' (in this specific sense) is a pure intransitive verb that does not permit objects of any kind.
