The second character of 깡패 is 牌 - which I thought meant 'signboard', 'card'. Arch Chinese gives the meaning as
cards, game pieces, signboard, placard, plate, tablet
How is this relevant to the meaning of the word 깡패 (gangster)?
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Learning Chinese letters could take you nowhere when you analyze Korean words seemingly originated from Chinese words. As a Korean language scholar, the late Mr. 정재도 asserted that when 패 means a group, 패 is relevant to the sound of 牌 not to the meaning of 牌. In my view, this is convincing, as such sound-borrowing had been a must until the Korean characters were created. If his statement is true, the Korean 패 is just a separate word from the Chinese 牌. The following is what he wrote about 패 and 牌:
우리말 ‘패’는 ‘동아리’, ‘무리’라는 뜻으로 ‘패거리, 남사당패, 노름패, 왈패 …’들처럼 ‘사람’에 관해서만 쓰인다. 한자말 ‘牌’는 ‘패찰, 군대 조직, 열 집’ 따위를 일컫는 말인데, 여러 가지 한자말에만 쓰인다. 우리말 ‘패’를 중국에서는 ‘牌’라 하지 않고, ‘파’(派)나, ‘훠’(人+火 果+多)라고 한다. 우리말 ‘패’와 한자말 ‘牌’는 다른 말이다 (link).
He mentioned here that 패 as in 패거리 (a gang/clique) is used only to describe someone or some people who belong to a group whereas 牌 that does not mean a group but a military organization, a train tablet, ten houses, etc. is in various words composed of Chinese letters only. He added that in China, 派, 伙, or 夥 is used to mean this 패.
On the other hand, a Korean book's author regards "a group of royal guards in the Joseon dynasty" as the origin of 패 (link). In addition, the Standard Korean Language Dictionary that 패 was used to call the smallest military group. However, they do not explain how 패 relates to 牌, although I have found that there was 방패군(防牌軍) once in the Joseon dynasty. 방패군 (or 방패 in short) was the military (group) armed with 방패 (shields) to block the enemies' attacks in the place which is now Seoul. But still, I am unsure whether 방패군 is directly related to 패 meaning "a group."
As for 깡패, the National Institute of the Korean Language provides the etymology information (link), based on 홍윤표's publication, "우리말 뿌리 찾기."
깡패 (영어의 gang에 ‘패거리’의 ‘패’가 결합된 말)
This means that 깡패 is a compound word of "gang" and "패거리."
Consider the following site : link I will summarize it
i) 굳셀(=strong)강 무리(=group)패 : That is, 강패 is changed into 깡패.
Further, there is an another origin : Gang in the word gangster is pronounced as 갱 or 깽
That is, 갱패 = gang + 무리 패, which is a union of English word and China character.
That is, 갱패 is changed into 깡패
ii) 종로패 = a group of gangsters in 종로 city
건달패 = a group of gangsters
풍물패 = a group of Korean traditional musicians
내 패거리가 도착하면 시작하자 Let's fight when my group of friends arrives
그들의 싸움은 양가의 팻싸움으로 커졌다 Their fight brings group's fighting between their families
a) 굳셀(=strong) 강 카드(=card) 패
b) 깡 = A desperate characteristic or strength
c) In baduk game, there is 패(우두머리=leader)싸움.
Not to be trustworthy, but the Standard Korean Language Dictionary explains it as “gang-牌.”
A suitable translation for the “牌” here would be “a posse” in the informal sense (or maybe in the formal sense).
No idea how it got the meaning of “a posse.” Maybe it was derived from the sense of “a nameplate” or “a shield,” I guess?