Their imam forgot a prostration then he repeated the rak’ah, but some of them did not repeat it
Question

                During ‘Isha’ prayer the imam led us in prayer, but he was not the official imam. In the last rak’ah he did not do the second prostration, and he said the salaam without anyone noticing it or reminding him. A few minutes later, one of the worshippers came and told him about that, so he got up straight away and said, “We will repeat the last rak’ah in order to do the prostration that was forgotten.” 

Is this correct? Is it was not correct, then what is the correct thing to do? What is the ruling on those who did not do the second prostration with them?.
Praise be to Allah.
      Firstly: 

    If the imam forgets something during the prayer, then the 
    Prophet  (peace and blessings of Allaah be upon him) commanded the 
    members of the congregation to remind him. He said: “I am only human like 
    you, and I forget as you forget, so if I forget then remind me.” Narrated by 
    al-Bukhaari, 401. So the congregation in the mosque should have said tasbeeh 
    (“Subhaan Allaah”) so that the imam would notice and would do the second 
    prostration. 

    Secondly: 

    Both the first and second prostrations are pillars or 
    essential parts of the prayer, without which the prayer is not valid. 
    Whoever deliberately omits one or both of them has sinned, and his prayer is 
    invalid. Whoever forgets one or both of them has to do what he forgot when 
    he remembers it, whether he is leading a congregation, following an imam or 
    praying on his own. Whoever does not do that, his prayer is invalid. 

    Shaykh Muhammad ibn Saalih al-‘Uthaymeen (may Allaah have 
    mercy on him) said: 

    The pillars or essential parts (of the prayer) are obligatory 
    and more important than the waajibat (obligatory parts of prayer), but they 
    differ from then in that the pillars are not waived if one forgets them 
    whereas the waajibaat are waived in that case and may be made up for by 
    doing the prostration of forgetfulness (sujood al-sahw), unlike the pillars. 
    Hence whoever forgets a pillar, his prayer is not valid without it. 

    Al-Sharh al-Mumti’, 3/315 

    And he said: 
The evidence that the 
    pillars cannot be made up for by the prostration of forgetfulness is the 
    fact that when the Prophet  (peace and blessings of Allaah be upon him) 
    said the salaam after praying two rak'ahs of Zuhr or ‘Asr, he completed it 
    and did what he had omitted, and did the prostration of forgetfulness. This 
    indicates that the pillars of prayer cannot be waived by doing the 
    prostration of forgetfulness, and they must be done. 

    Al-Sharh al-Mumti’, 3/323 

    With regard to what your imam did by repeating the final 
    rak’ah in full after being reminded, this is in accordance with one of the 
    two scholarly opinions on this matter, which is that the one who omits a 
    pillar or essential part of the final rak’ah and does not realize it until 
    after saying the tasleem should do the rak’ah in full. This is the view of 
    Imam Ahmad (may Allaah have mercy on him). See al-Mughni, 1/658. 
This view was favoured 
    by Shaykh Ibn Baaz (may Allaah have mercy on him), who was asked about an 
    imam who forget the final prostration in ‘Asr prayer. He stood up and 
    offered a complete rak’ah, then he said the tashahhud and the tasleem, then 
    he did the prostration of forgetfulness. He said: This is what is prescribed 
    in Islam. If the imam forgot a prostration and said the tasleem, then he 
    remembered it or it was pointed out to him, he should stand up and do a full 
    rak’ah, then say the salaam, then do the prostration of forgetfulness after 
    saying the salaam. This is better. The same ruling applies to one who is 
    praying alone. But if he does the prostration of forgetfulness before the 
    salaam, there is nothing wrong with that, but doing it after the salaam is 
    preferable. 

    Majmoo’ Fataawa Ibn Baaz, 
    11/277 

    The second view concerning this matter is that he does not 
    have to do the rak'ah in full, rather he should do the pillar that he forgot 
    and whatever comes after it. This is the view of Imam al-Shaafa’i (may 
    Allaah have mercy on him). See al-Majmoo’, 4/33. This view was 
    favoured by Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him). See 
    Sharh al-Mumti’, 3/374. 
The point is that the 
    prayer of the imam, and the prayer of those who completed the prayer with 
    him, is valid. 

    But for those who did not complete the prayer and did not do 
    the prostration that was omitted, their prayer is not valid, and they have to repeat it. 

    And Allaah knows best.