Groups and subgroups

Question

I have been told that {0, 2, 4, 6, 8} is a subgroup of the multiplicative integer mod 10. I know that the operation is multiplication, so I understand that every element has its inverse within the set, with respect to the operation. However, the part that i do not understand is why it is labeled as a subgroup when the identity of the multiplicative operation (1) is not in the set. Your help would be very much appreciated.

Answer 1

Your set is a subgroup of the additive group of integers modulo $10$, namely $\langle 2\rangle \lt \langle \mathbb Z, +\rangle$, but is not a subgroup of the multiplicative group of integers modulo $10$, because, as you state, the multiplicative identity $1$ is not listed, and because $0$ has no multiplicative inverse.

Answer 2

As @amWhy pointed out, 0 has no multiplicative inverse, so under multiplication this set under multiplication isn't a group at all. There would be a similar problem with trying to call this set a subgroup of the multiplicative integers mod 10: if "the multiplicative integers mod 10" means $\mathbb{Z}/10 \mathbb{Z}$ under multiplication, then it isn't a group, and hence can't have subgroups. If "the multiplicative integers mod 10" means the elements of $\mathbb{Z} / 10 \mathbb{Z}$ that are invertible under multiplication, then this is a group, but $S := \{0,2,4,6,8\}$ isn't even a subset of this group.

One thing to notice is that 6 is actually a multiplicative identity for $S$, so $S$ is a ring with 1. This is an example of the at-first surprising fact that if $R$ is a ring with 1 and $S \subseteq R$ that does not contain $1_R$, it is still possible for $S$ to be a ring with $1$, and in fact $1_S$ may even be a zero-divisor in $R$ (in this case $R = \mathbb{Z} / 10 \mathbb{Z}$).