# Find identity and inverses of the group $(\Bbb{Z}_6,+)\times(\Bbb{Z}_5,\cdot)$.

Let $$(\Bbb{Z}_6,+)$$ and $$(\Bbb{Z}_5,\cdot)$$ be two groups.

Define an operation for $$(\Bbb{Z}_6,+)\times(\Bbb{Z}_5,\cdot)$$ such that $$(\Bbb{Z}_6,+)\times(\Bbb{Z}_5,\cdot)$$ is a group. Find the identity and inverses of the product.

I know the following:

Property If $$(G_1,*_1)$$ and $$(G_2,*_2)$$ are two GROUPS, then the set $$G_1\times G_2$$ with the operation $$(a,b)*(c,d)=(a*_1c,b*_2d)$$ is also a GROUP.

But here, $$(\Bbb{Z}_5,\cdot)$$ is NOT a group since the inverse element of $$0$$ does not exist. So what should we do here?

**Should we consider another operation like $$(a,b)*(c,d)=(a+_6c,b+_5d)?$$

But the hypothesis is not true because $$(\Bbb{Z}_5,\cdot)$$ is not a group, so I do not know if I can define such operation of the product of "two" groups.

Any help?

• Perhaps the author of the question meant $\cdot$ to mean addition also, and since $+$ is taken, they chose $\cdot$. Besides, it even says "Let $(\Bbb Z_6,+)$ and $(\Bbb Z_5,\cdot )$ be two groups". – Shaun Nov 21 '19 at 0:29
• Ah, just realized your issue is the operation of $G$ is addition in the first component but multiplication in the second. You're probably fine once you realize this – oshill Nov 21 '19 at 0:29
• Yes $(\Bbb Z_5, \cdot)$ is not a group so we cannot define the product ...However $(\Bbb Z_5 \setminus \{0\}, \odot)$ is a group – Chinnapparaj R Nov 21 '19 at 0:29
• @Shaun I don't think so; the author defines (\cdot) as a multiplication (saw in other exercises; this is one of them). – manooooh Nov 21 '19 at 0:34
• @oshill yeah! The problem is in the second component! Because all the elements of the form $(x,0)$ (where $x\in\Bbb{Z}_6$) does not have inverse! – manooooh Nov 21 '19 at 0:35

Assuming $$(\Bbb Z_5,\cdot)=U(5)$$ is the group of units modulo five, the identity of $$G:=(\Bbb Z_6, +)\times U(5)$$ is $$(_6, _5)$$ and the inverse of $$([a]_6, [b]_5)$$ in $$G$$ is $$([-a]_6, [c]_5)$$, where $$c$$ is such that $$5x+bc=1$$ for some $$x\in \Bbb Z$$ and $$[d]_n:=\{k\in \Bbb Z\mid n \text{ divides } (k-d)\}.$$

Assuming $$(\Bbb Z_6,+)=(\Bbb Z_6, +_6)$$ and $$(\Bbb Z_5,\cdot)=(\Bbb Z_5, +_5)$$, then the identity of $$H:=(\Bbb Z_6, +_6)\times(\Bbb Z_5, +_5)$$ is $$(_6, _5)$$ and the inverse of $$([a]_6, [b]_5)$$ in $$H$$ is $$([-a]_6, [-b]_5)$$.

NB: Since five is prime, there is only one group of order five up to isomorphism. The question specifies that $$(\Bbb Z_5,\cdot)$$ is a group.

• Yes $\Bbb{Z}_{5}=\{0,1,2,3,4\}$. What does $5x+bc=1$ have to do with $[d]_n$? Thank you! – manooooh Nov 21 '19 at 0:44
• Well, $5x=1-bc$, so $5\mid 1-bc$. Hence $_5=[bc]_5=[b]_5\cdot [c]_5$, so $[b]_5$ has $[c]_5$ as an inverse. – Shaun Nov 21 '19 at 0:48
• But where do we put $[d]_n$? – manooooh Nov 21 '19 at 0:49
• No worries, @manooooh. Please don't forget to upvote & accept this answer if it works for you. – Shaun Nov 21 '19 at 0:54
• But with your definition of inverse I cannot figure what should be the inverse of $(0,0)$. It should be $(0,c)$, where $c$ is such that $5x+0c=5x=1$ for some $x\in\Bbb{Z}$. But $x=1/5\notin\Bbb{Z}$. So $(0,0)$ has no inverse. What am I doing wrong? – manooooh Nov 21 '19 at 0:54