This is a question from a practice paper which I have an exam on the first day back. I just wanted to run it by someone else to see if what I was doing was correct.

I structured my entire answer on the properties of a group:Which, if any, of the following subsets of $\displaystyle S_4$ are subgroups? Give brief reasons for your answers.

i). {i, (1 2)}

ii). {(1 2 3), (1 3 2)}

iii). {i, (1 2)(3 4), (1 4)(2 3)}

iv). {i, (1 2), (1 4), (2 4), (1 4 2), (1 2 4)}.

1). Closed

2). Associativity

3). The existence of a neutral element

4). The presence of inverses.

i). Is not associative since starting at 1, there is no path to 3 or 4 (The definition of associativity states that "$\displaystyle \forall g_1, g_2 \in G \exists g_3 \in G s.t g_1g_2=g_3$". Here the "$\displaystyle \forall g_1, g_2$" statement fails.).

All the other properties hold.

Therefore i) is not a subgroup of$\displaystyle S_4$.

ii). For this case, all the properties hold except for the presence of a neutral element. Starting at any of 1,2,3 or 4 there is no path that takes either of these numbers back to themselves without stopping at another number first.

Hence ii). is not a subgroup of $\displaystyle S_4$.

iii). All the properties hold for this set.

Therefore iii). is a subgroup of $\displaystyle S_4$.

iv). All the properties once again hold except for associativity. Starting from 1,2 or 4 it is impossible to get to 3.

Therefore iv). is not a subgroup.

I think I have done this correctly but my answers to i) and iv) are the same. There is also only one property for a subgroup failing each time.

I would appreciate it if someone could check these answers and see if it is what they would put.